Tootfinder

The Optimal Smoothings of Sublinear Functions and Convex Cones
Thabo Samakhoana, Benjamin Grimmer
https://arxiv.org/abs/2508.06681 https://arxiv.org/pdf/25…

The Optimal Smoothings of Sublinear Functions and Convex Cones
This paper considers the problem of smoothing convex functions and sets, seeking the nearest smooth convex function or set to a given one. For convex cones and sublinear functions, a full characterization of the set of all optimal smoothings is given. These provide if and only if characterizations of the set of optimal smoothings for any target level of smoothness. Optimal smoothings restricting to either inner or outer approximations also follow from our theory. Finally, we apply our theory to…

Gradient regularity for widely degenerate elliptic partial differential equations
Michael Strunk
https://arxiv.org/abs/2506.11708 https://

Gradient regularity for widely degenerate elliptic partial differential equations
In this paper, we investigate the regularity of weak solutions $u\colonΩ\to\mathbb{R}$ to elliptic equations of the type \begin{equation*} \mathrm{div}\, \nabla \mathcal{F}(x,Du) = f\qquad\text{in $Ω$}, \end{equation*} whose ellipticity degenerates in a fixed bounded and convex set $E\subset\mathbb{R}^n$ with $0\in \mathrm{Int}\, E$. Here, $Ω\subset\mathbb{R}^n$ denotes a bounded domain, and $\mathcal{F} \colon Ω\times\mathbb{R}^n \to\mathbb{R}_{\geq 0}$ is a function with the properties:…

A Diestel-Faires type result for multimeasures
Jos\'e Rodr\'iguez
https://arxiv.org/abs/2506.11872 https://arxiv.org/pdf/2506…

A Diestel-Faires type result for multimeasures
Let $X$ be a real Banach space and let $Y \subseteq X^*$ be a linear subspace having the Orlicz-Thomas property, that is, for each $σ$-algebra $Σ$ and for each map $ν:Σ\to X$, the countable additivity of the composition $x^*\circ ν$ for all $x^*\in Y$ implies the countable additivity of $ν$. We show that the Orlicz-Thomas property allows to test countable additivity of set-valued maps. Namely, if $M$ is a map defined on a $σ$-algebra $Σ$ whose values are convex, $σ(X,Y)$-compact, bound…

Controlled Reach-avoid Set Computation for Discrete-time Polynomial Systems via Convex Optimization
Taoran Wu, Yiling Xue, Dejin Ren, Arvind Easwaran, Martin Fr\"anzle, Bai Xue
https://arxiv.org/abs/2506.06679

Controlled Reach-avoid Set Computation for Discrete-time Polynomial Systems via Convex Optimization
This paper addresses the computation of controlled reach-avoid sets (CRASs) for discrete-time polynomial systems subject to control inputs. A CRAS is a set encompassing initial states from which there exist control inputs driving the system into a target set while avoiding unsafe sets. However, efficiently computing CRASs remains an open problem, especially for discrete-time systems. In this paper, we propose a novel framework for computing CRASs which takes advantage of a probabilistic perspec…

This https://arxiv.org/abs/2504.03575 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the maximal volume of empty convex bodies amidst multivariate dilates of a lacunary integer sequence
Let \((a_n)_{n \in \mathbb{N}}\) be a lacunary sequence of integers satisfying the Hadamard gap condition. For any fixed dimension $d \geq 1$, we establish asymptotic upper bounds for the maximal gap in the set of dilates \(\{\boldsymbolα a_n \}_{n \leq N}\) modulo 1 as $N \to \infty$, for Lebesgue--almost all dilation vectors $\boldsymbolα \in [0,1]^d$. More precisely, we prove that for any lacunary \((a_n)_{n \in \mathbb{N}}\) and Lebesgue--almost all $\boldsymbolα$, every convex set in $[…

This https://arxiv.org/abs/2306.07237 has been replaced.
link: https://scholar.google.com/scholar?q=a

Three Edge-disjoint Plane Spanning Paths in a Point Set
We consider the following problem: Given a set $S$ of $n$ distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on $S$? Each spanning path must be crossing-free, but edges from different paths are allowed to intersect at arbitrary points. It is known that if the points of $S$ are in convex position, then $\lfloor n/2 \rfloor$ such paths always exist. However, for general point sets, the best known construction yields only two edge-disjoint plane sp…

Billiards with singular invariant curves
Stefano Baranzini
https://arxiv.org/abs/2508.08990 https://arxiv.org/pdf/2508.08990…

Billiards with singular invariant curves
We investigate the regularity of invariant curves of rotation number $1/2$ for a special class of symplectic twist maps of the annulus, billiard maps. We construct strictly convex smooth tables close to the circle having singular (i.e. not $C^1$) invariant curves. Our method relies on a modification of the classical string construction and allows precise control over the location of singularities: they form a discrete set whose closure can contain virtually any closed subset of $\mathbb{S}^1$. …

Tightening the mixed integer linear formulation for the piecewise linear approximation in general dimensions
Quentin Ploussard, Xiang Li, Matija Pavi\v{c}evi\'c
https://arxiv.org/abs/2508.09395

Tightening the mixed integer linear formulation for the piecewise linear approximation in general dimensions
This paper addresses the problem of tightening the mixed-integer linear programming (MILP) formulation for continuous piecewise linear (CPWL) approximations of data sets in arbitrary dimensions. The MILP formulation leverages the difference-of-convex (DC) representation of CPWL functions. We introduce the concept of well-behaved CPWL interpolations and demonstrate that any CPWL interpolation of a data set has a well-behaved version. This result is critical to tighten the MILP problem. We presen…

On the singular set of the free boundary for a Monge-Amp\`ere obstacle problem
Tianling Jin, Xushan Tu, Jingang Xiong
https://arxiv.org/abs/2506.08387 http…

On the singular set of the free boundary for a Monge-Ampère obstacle problem
This is a continuation of our earlier work [14] on the Monge-Ampère obstacle problem \[ \det D^2 v = v^q χ_{\{v>0\}}, \quad v \geq 0 \text{ convex} \] with $q \in [0,n)$, where we studied the regularity of the strictly convex part of the free boundary. In this work, we examine the non-strictly convex part of the free boundary and establish optimal dimension bounds for its flat portion. Additionally, we investigate the strong maximum principle and a stability property for this Monge-Ampère …

Set-Based Control Barrier Functions and Safety Filters
Kim P. Wabersich, Felix Berkel, Felix Gruber, Sven Reimann
https://arxiv.org/abs/2507.07805 https://…

Set-Based Control Barrier Functions and Safety Filters
High performance and formal safety guarantees are common requirements for industrial control applications. Control barrier function (CBF) methods provide a systematic approach to the modularization of safety and performance. However, the design of such CBFs can be challenging, which limits their applicability to large-scale or data-driven systems. This paper introduces the concept of a set-based CBF for linear systems with convex constraints. By leveraging control invariant sets from reachabili…

A $\sigma$-morphic convex protoset
Aleksa D\v{z}uklevski
https://arxiv.org/abs/2507.20867 https://arxiv.org/pdf/2507.20867…

A $σ$-morphic convex protoset
We say that a tile is $σ$-morphic if it tiles the plane in exactly $\aleph_0$ many noncongruent ways (up to an isometry). It is an unsolved problem of whether a $σ$-morphic tile exist in the plane. In this note we present a construction of a set of convex tiles that is $σ$-morphic. The result is interesting since all the constructions of $σ$-morphic sets of tiles that arise in the literature make use of bumps and nicks, which necessarily make the tiles non-convex. We construct our set by cl…

Weak solutions of Stochastic Volterra Equations in convex domains with general kernels
Eduardo Abi Jaber, Aur\'elien Alfonsi, Guillaume Szulda
https://arxiv.org/abs/2506.04911

Weak solutions of Stochastic Volterra Equations in convex domains with general kernels
We establish new weak existence results for $d$-dimensional Stochastic Volterra Equations (SVEs) with continuous coefficients and possibly singular one-dimensional non-convolution kernels. These results are obtained by introducing an approximation scheme and showing its convergence. A particular emphasis is made on the stochastic invariance of the solution in a closed convex set. To do so, we extend the notion of kernels that preserve nonnegativity introduced in \cite{Alfonsi23} to non-convolut…

Minimal displacement set for CAT(0) cubical complexes
Ioana-Claudia Lazar
https://arxiv.org/abs/2505.23318 https://arxiv.org/pdf/2505…

Minimal displacement set for CAT(0) cubical complexes
We investigate the structure of the minimal displacement set in CAT(0) cubical complexes. We show that such set is convex, it is locally endowed with a CAT(0) metric and it is simply connected.

On compact sets possessing $q$-convex functions
Thomas Pawlaschyk, Nikolay Shcherbina
https://arxiv.org/abs/2505.24588 https://arxiv.…

On compact sets possessing $q$-convex functions
We show that there exists a $q$-convex function in a neighborhood of a compact set $K$ in a complex manifold $\mathcal{M}$ if and only if the $q$-nucleus of this compact set is empty. The latter can be characterized as the maximal $q$-pseudoconcave subset of $K$, i.e., a subset of $K$ containing all other compact $q$-pseudoconcave subsets in $K$.

An unconditional lower bound for the active-set method in convex quadratic maximization
Eleon Bach, Yann Disser, Sophie Huiberts, Nils Mosis
https://arxiv.org/abs/2507.16648

An unconditional lower bound for the active-set method in convex quadratic maximization
We prove that the active-set method needs an exponential number of iterations in the worst-case to maximize a convex quadratic function subject to linear constraints, regardless of the pivot rule used. This substantially improves over the best previously known lower bound [IPCO 2025], which needs objective functions of polynomial degrees $ω(\log d)$ in dimension $d$, to a bound using a convex polynomial of degree 2. In particular, our result firmly resolves the open question [IPCO 2025] of whe…

A strengthened bound on the number of states required to characterize maximum parsimony distance
Mareike Fischer, Steven Kelk, Sofia Vazquez Alferez
https://arxiv.org/abs/2506.09888

A strengthened bound on the number of states required to characterize maximum parsimony distance
In this article we prove that the distance $d_{\mathrm{MP}}(T_1,T_2) = k$ between two unrooted binary phylogenetic trees $T_1, T_2$ on the same set of taxa can be defined by a character that is convex on one of $T_1, T_2$ and which has at most $2k$ states. This significantly improves upon the previous bound of $7k-5$ states. We also show that for every $k \geq 1$ there exist two trees $T_1, T_2$ with $d_{\mathrm{MP}}(T_1,T_2) = k$ such that at least $k+1$ states are necessary in any character t…

Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter
Gianmarco Picarella, Marc van Kreveld, Frank Staals, Sjoerd de Vries
https://arxiv.org/abs/2507.04933

Computing Largest Subsets of Points Whose Convex Hulls have Bounded Area and Diameter
We study the problem of computing a convex region with bounded area and diameter that contains the maximum number of points from a given point set $P$. We show that this problem can be solved in $O(n^6k)$ time and $O(n^3k)$ space, where $n$ is the size of $P$ and $k$ is the maximum number of points in the found region. We experimentally compare this new algorithm with an existing algorithm that does the same but without the diameter constraint, which runs in $O(n^3k)$ time. For the new algorith…

Optimal $C^{\frac{1}{2}}$ regularity of the Boltzmann equation in non-convex domains
Gayoung An, Donghyun Lee
https://arxiv.org/abs/2507.07353 https://

Optimal $C^{\frac{1}{2}}$ regularity of the Boltzmann equation in non-convex domains
Regularity of the Boltzmann equation, particularly in the presence of physical boundary conditions, heavily relies on the geometry of the boundaries. In the case of non-convex domains with specular reflection boundary conditions, the problem remained outstanding until recently due to the severe singularity of billiard trajectories near the grazing set, where the trajectory map is not differentiable. This challenge was addressed in [32], where $C^{\frac{1}{2}-}_{x,v}$ Hölder regularity was prov…

This https://arxiv.org/abs/2503.16398 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

The global convergence time of stochastic gradient descent in non-convex landscapes: Sharp estimates via large deviations
In this paper, we examine the time it takes for stochastic gradient descent (SGD) to reach the global minimum of a general, non-convex loss function. We approach this question through the lens of randomly perturbed dynamical systems and large deviations theory, and we provide a tight characterization of the global convergence time of SGD via matching upper and lower bounds. These bounds are dominated by the most "costly" set of obstacles that the algorithm may need to overcome in order to reach…

Trajectory Optimization for UAV-Based Medical Delivery with Temporal Logic Constraints and Convex Feasible Set Collision Avoidance
Kaiyuan Chen, Yuhan Suo, Shaowei Cui, Yuanqing Xia, Wannian Liang, Shuo Wang
https://arxiv.org/abs/2506.06038

Trajectory Optimization for UAV-Based Medical Delivery with Temporal Logic Constraints and Convex Feasible Set Collision Avoidance
This paper addresses the problem of trajectory optimization for unmanned aerial vehicles (UAVs) performing time-sensitive medical deliveries in urban environments. Specifically, we consider a single UAV with 3 degree-of-freedom dynamics tasked with delivering blood packages to multiple hospitals, each with a predefined time window and priority. Mission objectives are encoded using Signal Temporal Logic (STL), enabling the formal specification of spatial-temporal constraints. To ensure safety, c…

Geometric implications of weak tiling
Mihail N. Kolountzakis, Nir Lev, M\'at\'e Matolcsi
https://arxiv.org/abs/2506.23631 https://

Geometric implications of weak tiling
The notion of weak tiling played a key role in the proof of Fuglede's spectral set conjecture for convex domains, due to the fact that every spectral set must weakly tile its complement. In this paper, we revisit the notion of weak tiling and establish some geometric properties of sets that weakly tile their complement. If $A \subset \mathbb{R}^d$ is a convex polytope, we give a direct and self-contained proof that $A$ must be symmetric and have symmetric facets. If $A \subset \mathbb{R}$ is a …

Extreme points of unital completely positive maps invariant under partial action
Chaitanya J. Kulkarni, Md Amir Hossain
https://arxiv.org/abs/2507.20797 https://

Extreme points of unital completely positive maps invariant under partial action
The classical Choquet theorem establishes a barycentric decomposition for elements in a compact convex subset of a locally convex topological vector space. This decomposition is achieved through a probability measure that is supported on the set of extreme points of the subset. In this work, we consider a partial action $τ$ of a group $G$ on a $C^\ast$-algebra $\mathcal{A}$. For a fixed Hilbert space $\mathcal{H}$, we consider the set of all unital completely positive maps from $\mathcal{A}$ t…

Dual realizations of Bergman spaces on strongly convex domains
Agniva Chatterjee
https://arxiv.org/abs/2506.02913 https://arxiv.org/p…

Dual realizations of Bergman spaces on strongly convex domains
The Fantappiè and Laplace transforms realize isomorphisms between analytic functionals supported on a convex compact set $K\subset{\mathbb C}^n$ and certain spaces of holomorphic functions associated with $K$. Viewing the Bergman space of a bounded domain in ${\mathbb C}^n$ as a subspace of the space of analytic functionals supported on its closure, the images of the restrictions of these transforms have been studied in the planar setting. For the Fantappiè transform, this was done for simply…

Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case
Aur\'elien Alfonsi, Benjamin Jourdain
https://arxiv.org/abs/2506.23981

Wasserstein projections in the convex order: regularity and characterization in the quadratic Gaussian case
In this paper, we first show continuity of both Wasserstein projections in the convex order when they are unique. We also check that, in arbitrary dimension $d$, the quadratic Wasserstein projection of a probability measure $μ$ on the set of probability measures dominated by $ν$ in the convex order is non-expansive in $μ$ and Hölder continuous with exponent $1/2$ in $ν$. When $μ$ and $ν$ are Gaussian, we check that this projection is Gaussian and also consider the quadratic Wasserstein p…

This https://arxiv.org/abs/2501.03645 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Big convex polytopes or rich hyperplanes
For natural numbers $n$ and $l > d \geq 2$, let $ES_d(l,n)$ be the minimum $N$ such that any set of at least $N$ points in $\mathbb{R}^d$ contains either $l$ points contained in a common $(d-1)$-dimensional hyperplane or $n$ points in convex position. In this paper, we give the upper and lower bounds for $ES_d(l,n)$.

Upgrading survival models with CARE
William G. Underwood, Henry W. J. Reeve, Oliver Y. Feng, Samuel A. Lambert, Bhramar Mukherjee, Richard J. Samworth
https://arxiv.org/abs/2506.23870

Upgrading survival models with CARE
Clinical risk prediction models are regularly updated as new data, often with additional covariates, become available. We propose CARE (Convex Aggregation of relative Risk Estimators) as a general approach for combining existing "external" estimators with a new data set in a time-to-event survival analysis setting. Our method initially employs the new data to fit a flexible family of reproducing kernel estimators via penalised partial likelihood maximisation. The final relative risk estimator i…

Discrete and Continuous Difference of Submodular Minimization
George Orfanides, Tim Hoheisel, Marwa El Halabi
https://arxiv.org/abs/2506.07952 https://

Discrete and Continuous Difference of Submodular Minimization
Submodular functions, defined on continuous or discrete domains, arise in numerous applications. We study the minimization of the difference of two submodular (DS) functions, over both domains, extending prior work restricted to set functions. We show that all functions on discrete domains and all smooth functions on continuous domains are DS. For discrete domains, we observe that DS minimization is equivalent to minimizing the difference of two convex (DC) functions, as in the set function cas…

Decremental Greedy Polygons and Polyhedra Without Sharp Angles
David Eppstein
https://arxiv.org/abs/2507.04538 https://arxiv.org/pdf/…

Decremental Greedy Polygons and Polyhedra Without Sharp Angles
We show that the max-min-angle polygon in a planar point set can be found in time $O(n\log n)$ and a max-min-solid-angle convex polyhedron in a three-dimensional point set can be found in time $O(n^2)$. We also study the maxmin-angle polygonal curve in 3d, which we show to be $\mathsf{NP}$-hard to find if repetitions are forbidden but can be found in near-cubic time if repeated vertices or line segments are allowed, by reducing the problem to finding a bottleneck cycle in a graph. We formalize …

Piecewise Deterministic Sampling for Constrained Distributions
Jo\"el Tatang Demano, Paul Dobson, Konstantinos Zygalakis
https://arxiv.org/abs/2508.05462 https://

Piecewise Deterministic Sampling for Constrained Distributions
In this paper, we propose a novel class of Piecewise Deterministic Markov Processes (PDMP) that are designed to sample from constrained probability distributions $π$ supported on a convex set $\mathcal{M}$. This class of PDMPs adapts the concept of a mirror map from convex optimisation to address sampling problems. Such samplers provides unbiased algorithms that respect the constraints and, moreover, allow for exact subsampling. We demonstrate the advantages of these algorithms on a range of c…

This https://arxiv.org/abs/2503.10126 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_ees…

An LiGME Regularizer of Designated Isolated Minimizers -- An Application to Discrete-Valued Signal Estimation
For a regularized least squares estimation of discrete-valued signals, we propose a Linearly involved Generalized Moreau Enhanced (LiGME) regularizer, as a nonconvex regularizer, of designated isolated minimizers. The proposed regularizer is designed as a Generalized Moreau Enhancement (GME) of the so-called sum-of-absolute-values (SOAV) convex regularizer. Every candidate vector in the discrete-valued set is aimed to be assigned to an isolated local minimizer of the proposed regularizer while …

Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra
Eric Evert, Benjamin Passer
https://arxiv.org/abs/2507.20325 https://a…

Matrix convex sets over the Euclidean ball and polar duals of real free spectrahedra
We show that the free spectrahedron determined by universal anticommuting self-adjoint unitaries is not equal to the minimal matrix convex set over the ball in dimension three or higher. This example, as well as other matrix convex sets over the ball, then provides context for structure results on the extreme points of coordinate projections. In particular, we show that the free polar dual of a real free spectrahedron is rarely the projection of a real free spectrahedron, contrasting a prior re…

Biobjective optimization with M-convex functions
Ellen H. Fukuda, Satoru Iwata, Itsuki Nakagawa
https://arxiv.org/abs/2507.23423 https://arxiv.org/pdf/2507…

Biobjective optimization with M-convex functions
In this paper, we deal with two ingredients that, as far as we know, have not been combined until now: multiobjective optimization and discrete convex analysis. First, we show that the entire Pareto optimal value set can be obtained in polynomial time for biobjective optimization problems with discrete convex functions, in particular, involving an M$^\natural$-convex function and a linear function with binary coefficients. We also observe that a more efficient algorithm can be obtained in the s…

A Linear Time Algorithm for Finding Minimum Flip Sequences between Plane Spanning Paths in Convex Point Sets
Oswin Aichholzer, Joseph Dorfer
https://arxiv.org/abs/2507.02740

A Linear Time Algorithm for Finding Minimum Flip Sequences between Plane Spanning Paths in Convex Point Sets
We provide a linear time algorithm to determine the flip distance between two plane spanning paths on a point set in convex position. At the same time, we show that the happy edge property does not hold in this setting. This has to be seen in contrast to several results for reconfiguration problems where the absence of the happy edge property implies algorithmic hardness of the flip distance problem. Further, we show that our algorithm can be adapted for (1) compatible flips (2) local flips and…

Robust Voting under Uncertainty
Satoshi Nakada, Shmuel Nitzan, Takashi Ui
https://arxiv.org/abs/2507.22655 https://arxiv.org/pdf/2507.22655

Robust Voting under Uncertainty
This paper proposes normative criteria for voting rules under uncertainty about individual preferences. The criteria emphasize the importance of responsiveness, i.e., the probability that the social outcome coincides with the realized individual preferences. Given a convex set of probability distributions of preferences, denoted by $P$, a voting rule is said to be $P$-robust if, for each probability distribution in $P$, at least one individual's responsiveness exceeds one-half. Our main result …

This https://arxiv.org/abs/2502.10382 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On creating convexity in high dimensions
Given a subset $A$ of $\mathbb{R}^n$, we define \begin{align*} \mathrm{conv}_k(A) := \left\{ λ_1 s_1 + \cdots + λ_k s_k : λ_i \in [0,1], \sum_{i=1}^k λ_i = 1 , s_i \in A \right\} \end{align*} to be the set of vectors in $\mathbb{R}^n$ that can be written as a $k$-fold convex combination of vectors in $A$. Let $γ_n$ denote the standard Gaussian measure on $\mathbb{R}^n$. We show that for every $\varepsilon > 0$, there exists a subset $A$ of $\mathbb{R}^n$ with Gaussian measure $γ_n(A) \geq…

This https://arxiv.org/abs/2410.05132 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

On the uniqueness of tangent cones to area minimizing currents at boundaries with arbitrary multiplicity
We consider an area minimizing current $T$ in a $C^2$ submanifold $Σ$ of $\mathbb{R}^{m+n}$, with arbitrary integer boundary multiplicity $\partial T = Q [\![ Γ]\!]$ where $Γ$ is a $C^2$ submanifold of $Σ$. We show that at every density $Q/2$ boundary point the tangent cone to $T$ is unique and there is a power rate of convergence to the unique tangent cone. In particular, if $Γ$ is a closed manifold which lies at the boundary of a uniformly convex set $Ω$ and $Σ=\mathbb{R}^{m+n}$ then…

Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers
Vitali Pirau, Roland Hildebrand
https://arxiv.org/abs/2507.01812

Numerical analysis of the convex relaxation of the barrier parameter functional of self-concordant barriers
Self-concordant barriers are essential for interior-point algorithms in conic programming. To speed up the convergence it is of interest to find a barrier with the lowest possible parameter for a given cone. The barrier parameter is a non-convex function on the set of self-concordant barriers on a given cone, and finding an optimal barrier amounts to solving a non-convex infinite-dimensional optimization problem. In this work we study the degradation of the optimal value of the problem when the…

On Strongly Convex Sets and Farthest Distance Functions
Juan Enrique Mart\'inez-Legaz
https://arxiv.org/abs/2507.15053 https://ar…

On Strongly Convex Sets and Farthest Distance Functions
A polarity notion for sets in a Banach space is introduced in such a way that the second polar of a set coincides with the smallest strongly convex set with respect to R that contains it. Strongly convex sets are characterized in terms of their associated farthest distance functions, and farthest distance functions associated with strongly convex sets are characterized, too.

A Mean-Field Theory of $\Theta$-Expectations
Qian Qi
https://arxiv.org/abs/2507.22577 https://arxiv.org/pdf/2507.22577

A Mean-Field Theory of $Θ$-Expectations
The canonical theory of sublinear expectations, a foundation of stochastic calculus under ambiguity, is insensitive to the non-convex geometry of primitive uncertainty models. This paper develops a new stochastic calculus for a structured class of such non-convex models. We introduce a class of fully coupled Mean-Field Forward-Backward Stochastic Differential Equations where the BSDE driver is defined by a pointwise maximization over a law-dependent, non-convex set. Mathematical tractability is…

How Hard is it to be a Star? Convex Geometry and the Real Hierarchy
Marcus Schaefer, Daniel \v{S}tefankovi\v{c}
https://arxiv.org/abs/2506.18818 https://…

How Hard is it to be a Star? Convex Geometry and the Real Hierarchy
A set is star-shaped if there is a point in the set that can see every other point in the set in the sense that the line-segment connecting the points lies within the set. We show that testing whether a non-empty compact smooth region is star-shaped is $\forall\mathbb{R}$-complete. Since the obvious definition of star-shapedness has logical form $\exists\forall$, this is a somewhat surprising result, based on Krasnosel'ski\uı's theorem from convex geometry; we study several related complexity …

Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality
Vo Si Trong Long, Nguyen Mau Nam, Len White
https://arxiv.org/abs/2507.07377

Qualitative and Generalized Differentiation Properties of Optimal Value Functions with Applications to Duality
In this paper, we investigate general and generalized differentiation properties of theoptimal value function associated with perturbed optimization problems. We begin with a comprehensive analysis of its effective domain, epigraph, strict epigraph, convexity, near convexity, continuity, and Lipschitz-type behavior, in both convex and nonconvex settings. Next, we propose a duality framework for constrained optimization problems with set-valued constraints, based on the notion of the Fenchel con…

Partial regularity of optimal transport with Coulomb cost
Gero Friesecke, Tobias Ried
https://arxiv.org/abs/2508.01756 https://arxiv.org/pdf/2508.01756

Partial regularity of optimal transport with Coulomb cost
We prove that for two-marginal optimal transport with Coulomb cost, the optimal map is a $C^{1,α}$ diffeomorphism outside a closed set of Lebesgue measure zero provided the marginals are $α$-Hölder continuous and bounded away from zero and infinity. Excluding a set of measure zero is necessary as optimal maps for the Coulomb cost have long been known to exhibit jump singularities across codimension $1$ surfaces (even for smooth marginals on convex domains).

This https://arxiv.org/abs/2504.02611 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_csCG_…

Instance-Optimal Imprecise Convex Hull
Imprecise measurements of a point set P = (p1, ..., pn) can be modelled by a family of regions F = (R1, ..., Rn), where each imprecise region Ri contains a unique point pi. A retrieval models an accurate measurement by replacing an imprecise region Ri with its corresponding point pi. We construct the convex hull of an imprecise point set in the plane, where regions in F may be retrieved at unit cost. The goal is to determine the cyclic ordering of the convex hull vertices of P as efficiently as…

A Theory of $\theta$-Expectations
Qian Qi
https://arxiv.org/abs/2507.20353 https://arxiv.org/pdf/2507.20353

A Theory of $θ$-Expectations
The canonical theory of stochastic calculus under ambiguity, founded on sub-additivity, is insensitive to non-convex uncertainty structures, leading to an identifiability impasse. This paper develops a mathematical framework for an identifiable calculus sensitive to non-convex geometry. We introduce the $θ$-BSDE, a class of backward stochastic differential equations where the driver is determined by a pointwise maximization over a primitive, possibly non-convex, uncertainty set. The system's t…

Frank-Wolfe algorithm for star-convex functions
R. Diaz Millan, Orizon Pereira Ferreira, Julien Ugon
https://arxiv.org/abs/2507.17272 https://arxiv.org/pdf…

Frank-Wolfe algorithm for star-convex functions
We study the Frank-Wolfe algorithm for minimizing a differentiable function with Lipschitz continuous gradient over a compact convex set. To extend classical complexity bounds to certain non-convex functions, we focus on the class of \emph{star-convex functions}, which retain essential geometric properties despite the lack of convexity. We establish iteration-complexity bounds of $\mathcal{O}(1/k)$ for both the objective values and the duality gap under star-convexity, using diminishing, Armijo…

On Steiner entire function
Maria Dospolova, Mikhail Germanskov, Dmitry Zaporozhets
https://arxiv.org/abs/2507.11626 https://arxiv.org…

On Steiner entire function
We introduce and study the Steiner entire function, an analytic generating function for the intrinsic volumes of a convex compact set in a Hilbert space. This function extends the classical Steiner polynomial to infinite dimensions and encodes key geometric information about the set. We establish fundamental results on its order, type, and canonical product representation, and show how its analytic growth properties characterize Gaussian continuity. In particular, we provide new criteria for th…

On a general class of free boundary Monge-Amp\`ere equations
Tristan C. Collins, Benjy Firester
https://arxiv.org/abs/2508.05551 https://arxiv.org/pdf/2508…

On a general class of free boundary Monge-Ampère equations
We solve a general class of free boundary Monge-Ampère equations given by \[ \det D^2u = λ\dfrac{f(-u)}{g(u^\star)h(\nabla u)}χ_{\{u<0\}} \; \text{ in } \mathbb{R}^n, \quad \nabla u (\mathbb{R}^n) = P \] where $P$ is a bounded convex set containing the origin, and $h>0$ on $P$. We consider applications to optimal transport with degenerate densities, Monge-Ampère eigenvalue problems, and geometric problems including a hemispherical Minkowski problem and free boundary Kähler-Ricci solitons…

Odd-indexed Fibonacci numbers via pattern-avoiding permutations
Juan B. Gil, Felix H. Xu, William Y. Zhu
https://arxiv.org/abs/2506.15800 https://

Odd-indexed Fibonacci numbers via pattern-avoiding permutations
In this paper, we consider several combinatorial problems whose enumeration leads to the odd-indexed Fibonacci numbers, including certain types of Dyck paths, block fountains, directed column-convex polyominoes, and set partitions with no crossings and no nestings. Our goal is to provide bijective maps to pattern-avoiding permutations and derive generating functions that track certain positional statistics at the permutation level.

This https://arxiv.org/abs/2408.03218 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Limit theorems for the number of crossings and stress in projections of the random geometric graph
We consider the number of edge crossings in a random graph drawing generated by projecting a random geometric graph on some compact convex set $W\subset \mathbb{R}^d$, $d\geq 3$, onto a plane. The positions of these crossings form the support of a point process. We show that if the expected number of crossings converges to a positive but finite value, this point process converges to a Poisson point process in the Kantorovich-Rubinstein distance. We further show a multivariate central limit theo…

This https://arxiv.org/abs/2410.22038 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

A Cramér-Wold theorem for mixtures
We show how a Cramér-Wold theorem for a family of multivariate probability distributions can be used to generate a similar theorem for mixtures (convex combinations) of distributions drawn from the same family. Using this abstract result, we establish a Cramér-Wold theorem for mixtures of multivariate Gaussian distributions. According to this theorem, two such mixtures can be distinguished by projecting them onto a certain predetermined finite set of lines, the number of lines depending onl…

CLARSTA: A random subspace trust-region algorithm for convex-constrained derivative-free optimization
Yiwen Chen, Warren Hare, Amy Wiebe
https://arxiv.org/abs/2506.20335

CLARSTA: A random subspace trust-region algorithm for convex-constrained derivative-free optimization
This paper proposes a random subspace trust-region algorithm for general convex-constrained derivative-free optimization (DFO) problems. Similar to previous random subspace DFO methods, the convergence of our algorithm requires a certain accuracy of models and a certain quality of subspaces. For model accuracy, we define a new class of models that is only required to provide reasonable accuracy on the projection of the constraint set onto the subspace. We provide a new geometry measure to make …

Fast and Accurate Reconstruction of Voronoi Generators in Large Tessellations
Carlos M Hernandez-Suarez
https://arxiv.org/abs/2506.19076 https://

Fast and Accurate Reconstruction of Voronoi Generators in Large Tessellations
A Voronoi diagram partitions the plane into convex cells, each containing the points closest to a single generator. Given such a tessellation, the inverse Voronoi problem seeks the generator set \( S \) that produced it. Our algorithm selects a single interior cell with \( k \) edges and solves a compact, consistent linear system with \( 2(k+1) \) unknowns and \( 4k \) scalar equations to recover that cell's generator together with the \( k \) generators of its neighbors in one step. The remain…

Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups
E. A. Ladeishchikov, L. V. Lokutsievskiy, N. V. Prilepin
https://arxiv.org/abs/2507.00250

Explicit formulas for extremals in sub-Lorentzian and Finsler problems on 2- and 3-dimensional Lie groups
In this paper, we consider the problem of finding geodesics in a series of left-invariant problems endowed with sub-Lorentzian and Finsler structures. Explicit formulas for extremals are obtained in terms of convex trigonometric functions. In the sub-Lorentzian setting, the new trigonometric functions $\cosh_Ω$ and $\sinh_Ω$, developed here, prove especially useful; they generalize the classical $\cosh$ and $\sinh$ to the case of an unbounded convex set $Ω\subset\mathbb{R}^2$.

This https://arxiv.org/abs/2409.06512 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

Manifolds of absolutely continuous functions with values in an infinite-dimensional manifold and regularity properties of half-Lie groups
For $p\in [1,\infty]$, we define a smooth manifold structure on the set $AC_{L^p}([a,b],N)$ of absolutely continuous functions $γ\colon [a,b]\to N$ with $L^p$-derivatives for all real numbers $a

No-dimensional Tverberg-type problems
Alexander Polyanskii
https://arxiv.org/abs/2506.13451 https://arxiv.org/pdf/2506.13451

No-dimensional Tverberg-type problems
Recently, Adiprasito et al. have initiated the study of the so-called no-dimensional Tverberg problem. This problem can be informally stated as follows: Given $n\geq k$, partition an $n$-point set in Euclidean space into $k$ parts such that their convex hulls intersect a ball of relatively small radius. In this survey, we aim to present the recent progress towards solving the no-dimensional Tverberg problem and new open questions arising in its context. Also, we discuss the colorful variation…

Replaced article(s) found for cs.CG. https://arxiv.org/list/cs.CG/new
[1/1]:
- Convex Covering Using Collections of Convex Polygons and Set Cover
Guilherme D. da Fonseca

Maximal entropy in the moment body
Didier Henrion (LAAS-POP)
https://arxiv.org/abs/2507.02461 https://arxiv.org/pdf/2507.02461…

Maximal entropy in the moment body
A moment body is a linear projection of the spectraplex, the convex set of trace-one positive semidefinite matrices. Determining whether a given point lies within a given moment body is a problem with numerous applications in quantum state estimation or polynomial optimization. This moment body membership oracle can be addressed with semidefinite programming, for which several off-the-shelf interior-point solvers are available. In this paper, inspired by techniques from quantum information theo…

The matrix weighted real-analytic double fibration transforms
Hiroyuki Chihara, Shubham R. Jathar, Jesse Railo
https://arxiv.org/abs/2506.24067 https://

The matrix weighted real-analytic double fibration transforms
We show that the real-analytic matrix-weighted double fibration transform determines the analytic wavefront set of a vector-valued function. We apply this result to show that the matrix weighted ray transform is injective on a two-dimensional, non-trapping, real-analytic Riemannian manifold with strictly convex boundary. Additionally, we show that a real-analytic Higgs field can be uniquely determined from the nonabelian ray transform on real-analytic Riemannian manifolds of any dimension with …

Learning Acceleration Algorithms for Fast Parametric Convex Optimization with Certified Robustness
Rajiv Sambharya, Jinho Bok, Nikolai Matni, George Pappas
https://arxiv.org/abs/2507.16264

Learning Acceleration Algorithms for Fast Parametric Convex Optimization with Certified Robustness
We develop a machine-learning framework to learn hyperparameter sequences for accelerated first-order methods (e.g., the step size and momentum sequences in accelerated gradient descent) to quickly solve parametric convex optimization problems with certified robustness. We obtain a strong form of robustness guarantee -- certification of worst-case performance over all parameters within a set after a given number of iterations -- through regularization-based training. The regularization term is …

Implicit Constraint-Aware Off-Policy Correction for Offline Reinforcement Learning
Ali Baheri
https://arxiv.org/abs/2506.14058 https://

Implicit Constraint-Aware Off-Policy Correction for Offline Reinforcement Learning
Offline reinforcement learning promises policy improvement from logged interaction data alone, yet state-of-the-art algorithms remain vulnerable to value over-estimation and to violations of domain knowledge such as monotonicity or smoothness. We introduce implicit constraint-aware off-policy correction, a framework that embeds structural priors directly inside every Bellman update. The key idea is to compose the optimal Bellman operator with a proximal projection on a convex constraint set, wh…

A polynomial projective algorithm for convex feasibility problems with positive-definite constraints
Sergei Chubanov
https://arxiv.org/abs/2506.15484 https…

A polynomial projective algorithm for convex feasibility problems with positive-definite constraints
We study a class of projective transformations of spectraplexes associated with self-dual cones and, on this basis, propose a polynomial-time algorithm for convex feasibility problems with positive definite constraints. At each iteration of the algorithm, either a feasible solution is found or a suitable valid inequality inducing a projective transformation allowing to bring the solution set closer to the center of an associated spectraplex. The closeness to the center is measured in terms of a…

An augmented Lagrangian method for strongly regular minimizers in a class of convex composite optimization problems
Chengjing Wang, Peipei Tang
https://arxiv.org/abs/2507.12040

An augmented Lagrangian method for strongly regular minimizers in a class of convex composite optimization problems
In this paper, we study a class of convex composite optimization problems. We begin by characterizing the equivalence between the primal/dual strong second-order sufficient condition and the dual/primal nondegeneracy condition. Building on this foundation, we derive a specific set of equivalent conditions for the perturbation analysis of the problem. Furthermore, we employ the augmented Lagrangian method (ALM) to solve the problem and provide theoretical guarantees for its performance. Specific…

Dynamic Optimal Transport with optimal star shaped graphs
Marcello Carioni, Juliane Krautz, Jan-F. Pietschmann
https://arxiv.org/abs/2506.15007 https://

Dynamic Optimal Transport with optimal star shaped graphs
We study an optimal transport problem in a compact convex set $Ω\subset\mathbb{R}^d$ where bulk transport is coupled to dynamic optimal transport on a metric graph $ \mathsf{G} = (\mathsf{V},\mathsf{E})$ which is embedded in $Ω$. We prove existence of solutions for fixed graphs. Next, we consider varying graphs, yet only for the case of star-shaped ones. Here, the action functional is augmented by an additional penalty that prevents the edges of the graph to overlap. This allows to preserve t…

This https://arxiv.org/abs/2406.10065 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An Extended Validity Domain for Constraint Learning
We consider embedding a predictive machine-learning model within a prescriptive optimization problem. In this setting, called constraint learning, we study the concept of a validity domain, i.e., a constraint added to the feasible set, which keeps the optimization close to the training data, thus helping to ensure that the computed optimal solution exhibits less prediction error. In particular, we propose a new validity domain which uses a standard convex-hull idea but in an extended space. We …

This https://arxiv.org/abs/2406.10065 has been replaced.
initial toot: https://mastoxiv.page/@arXiv_mat…

An Extended Validity Domain for Constraint Learning
We consider embedding a predictive machine-learning model within a prescriptive optimization problem. In this setting, called constraint learning, we study the concept of a validity domain, i.e., a constraint added to the feasible set, which keeps the optimization close to the training data, thus helping to ensure that the computed optimal solution exhibits less prediction error. In particular, we propose a new validity domain which uses a standard convex-hull idea but in an extended space. We …

Spectral Methods for Polynomial Optimization
Elvira Moreno, Venkat Chandrasekaran
https://arxiv.org/abs/2507.16272 https://arxiv.org/…

Spectral Methods for Polynomial Optimization
We present a hierarchy of tractable relaxations to obtain lower bounds on the minimum value of a polynomial over a constraint set defined by polynomial equations. In contrast to previous convex relaxation techniques for this problem, our method is based on computing the smallest generalized eigenvalue of a pair of matrices derived from the problem data, which can be accomplished for large problem instances using off-the-shelf software. We characterize the algebraic structure in a problem that f…

Distributed Nash Equilibrium Seeking in Non-Monotone Games over the Simplex
Tatiana Tatarenko, S. Rasoul Etesami
https://arxiv.org/abs/2507.13244 https://

Distributed Nash Equilibrium Seeking in Non-Monotone Games over the Simplex
In this work, we present a novel characterization of approximate Nash equilibria in a class of convex games over the simplex. To achieve this, we regularize the utility functions using the Shannon entropy term, connect the solutions to the regularized game with the set of Nash equilibria, and formulate a multi-objective optimization problem to solve the regularized game. Based on the obtained properties of the stationary points in this optimization problem, we formulate two distributed heuristi…