Compact Convex Research Articles

Bi-Halfspace and Convex Hull Theorems for Translating Solitons

AbstractWhile it is well known from examples that no interesting “halfspace theorem” holds for properly immersed $n$-dimensional self-translating mean curvature flow solitons in Euclidean space $\mathbb {R}^{n+1}$, we show that they must all obey a general “bi-halfspace theorem” (aka “wedge theorem”): two transverse vertical halfspaces can never contain the same such hypersurface. The same holds for any infinite end. The proofs avoid the typical methods of nonlinear barrier construction for the approach via distance functions and the Omori–Yau maximum principle. As an application we classify the closed convex hulls of all properly immersed (possibly with compact boundary) $n$-dimensional mean curvature flow self-translating solitons $\Sigma ^n$ in ${\mathbb {R}}^{n+1}$ up to an orthogonal projection in the direction of translation. This list is short, coinciding with the one given by Hoffman–Meeks in 1989 for minimal submanifolds: all of ${\mathbb {R}}^{n}$, halfspaces, slabs, hyperplanes, and convex compacts in ${\mathbb {R}}^{n}$.

Pursuit Differential Game of Many Pursuers with Integral Constraints on Compact Convex Set

We study a pursuit differential game of many pursuers and one evader in the plane. We are given a compact convex subset of $${\mathbb {R}}^2$$, and the pursuers and evader move in this set. They cannot leave this set during the game. Control functions of players are subject to coordinate-wise integral constraints. If the state of a pursuer coincides with that of evader at some time, then we say that pursuit is completed. Pursuers try to complete the pursuit, and the aim of the evader is opposite. We obtain some conditions under which pursuit can be completed from any positions of the players in the given set. Moreover, we construct strategies for the pursuers. Also, we prove some properties of convex sets.

A condition that implies full homotopical complexity of orbits for surface homeomorphisms

We consider closed orientable surfaces $S$ of genus $g>1$ and homeomorphisms $f:S\rightarrow S$ isotopic to the identity. A set of hypotheses is presented, called a fully essential system of curves $\mathscr{C}$ and it is shown that under these hypotheses, the natural lift of $f$ to the universal cover of $S$ (the Poincaré disk $\mathbb{D}$), denoted by $\widetilde{f},$ has complicated and rich dynamics. In this context, we generalize results that hold for homeomorphisms of the torus isotopic to the identity when their rotation sets contain zero in the interior. In particular, for $C^{1+\unicode[STIX]{x1D716}}$ diffeomorphisms, we show the existence of rotational horseshoes having non-trivial displacements in every homotopical direction. As a consequence, we found that the homological rotation set of such an $f$ is a compact convex subset of $\mathbb{R}^{2g}$ with maximal dimension and all points in its interior are realized by compact $f$-invariant sets and by periodic orbits in the rational case. Also, $f$ has uniformly bounded displacement with respect to rotation vectors in the boundary of the rotation set. This implies, in case where $f$ is area preserving, that the rotation vector of Lebesgue measure belongs to the interior of the rotation set.

The minimum principle for affine functions and isomorphisms of continuous affine function spaces

Let X be a compact convex set and let ext X stand for the set of extreme points of X. We show that if $$f:X\rightarrow {\mathbb {R}}$$ is an affine function with the point of continuity property such that $$f\le 0$$ on $${\text {ext}}\,X$$, then $$f\le 0$$ on X. As a corollary of this minimum principle, we obtain a generalization of a theorem by C.H. Chu and H.B. Cohen by proving the following result. Let X, Y be compact convex sets such that every extreme point of X and Y is a weak peak point and let $$T:\mathfrak {A}^c(X)\rightarrow \mathfrak {A}^c(Y)$$ be an isomorphism such that $$\left\| T\right\| \cdot \left\| T^{-1}\right\| <2$$. Then $${\text {ext}}\,X$$ is homeomorphic to $${\text {ext}}\,Y$$.

A Variation Problem for Stress–Energy Tensor

In this paper, we introduce a functional $$\Phi _S$$ related to the stress–energy tensor of a smooth map between two Riemannian manifolds. We derive the first variation formula and the second variation formula of $$\Phi _S$$. We also use the stress–energy tensor to obtain some Liouville type results for some special maps. Finally, we obtain that the maps from or into the compact convex hypersurfaces $$M^m\ (m\ge 5)$$ of $$R^{m+1}$$ which are stable stress–energy stationary maps and satisfy the inequality $$3\lambda _m<\sum _{i=1}^{m-1}\lambda _i$$ must be constant.

On Lau’s conjecture

In 1976 during a conference in Halifax, A. Lau raised the question on whether left amenability property for a semitopological semigroup is sufficient to ensure the existence of a common fixed point for every jointly weak* continuous norm nonexpansive action on a nonempty weak* compact convex set in a dual Banach space. In this paper we establish a positive answer for discrete semigroups and for strongly left amenable semitopological semigroups.

Online Convex Optimization for Sequential Decision Processes and Extensive-Form Games

Regret minimization is a powerful tool for solving large-scale extensive-form games. State-of-the-art methods rely on minimizing regret locally at each decision point. In this work we derive a new framework for regret minimization on sequential decision problems and extensive-form games with general compact convex sets at each decision point and general convex losses, as opposed to prior work which has been for simplex decision points and linear losses. We call our framework laminar regret decomposition. It generalizes the CFR algorithm to this more general setting. Furthermore, our framework enables a new proof of CFR even in the known setting, which is derived from a perspective of decomposing polytope regret, thereby leading to an arguably simpler interpretation of the algorithm. Our generalization to convex compact sets and convex losses allows us to develop new algorithms for several problems: regularized sequential decision making, regularized Nash equilibria in zero-sum extensive-form games, and computing approximate extensive-form perfect equilibria. Our generalization also leads to the first regret-minimization algorithm for computing reduced-normal-form quantal response equilibria based on minimizing local regrets. Experiments show that our framework leads to algorithms that scale at a rate comparable to the fastest variants of counterfactual regret minimization for computing Nash equilibrium, and therefore our approach leads to the first algorithm for computing quantal response equilibria in extremely large games. Our algorithms for (quadratically) regularized equilibrium finding are orders of magnitude faster than the fastest algorithms for Nash equilibrium finding; this suggests regret-minimization algorithms based on decreasing regularization for Nash equilibrium finding as future work. Finally we show that our framework enables a new kind of scalable opponent exploitation approach.

Complexity of Methods for Approximating Convex Compact Bodies by Double Description Polytopes and Complexity Bounds for a Hyperball

A comparative analysis of the complexity of approaches to the approximation of convex compact bodies by double description polytopes is provided as applied to a ball. A complexity bound for the Estimate Refinement method is obtained in the case of approximation of a multidimensional ball.

Numerical Algorithm for Minimizing a Convex Function on the Intersection of a Smooth Surface and a Convex Compact Set

Numerical Algorithm for Minimizing a Convex Function on the Intersection of a Smooth Surface and a Convex Compact Set

Non-differentiable Vector Programming Problem and Duality for Higher-Order Cone Convex Functions

In this paper, non-differentiable fractional program containing support function of a compact convex set is studied. A class of higher-order cone convex function is introduced which not necessarily lies in any of the existing classes. A non-trivial example of such type of functions is also introduced. Then Schaible dual is formulated for the above-said program for which weak and strong duality theorems are stated and proven using the newly defined function.

Remarks on random countable stable zonotopes

In the paper we study random stable countable zonotopes in Rd, given by Zα=⨁k=1∞Γk−1α[0,εk],0<α<1, where ⊕ stands for Minkowski sum, Γk=∑1kτj, {τj,j≥1} are i.i.d. random variables with common standard exponential distribution, and {εk,k≥1} are i.i.d. random vectors in Rd with common distribution σ, concentrated on the unit sphere of Rd. The sequences (τk) and (εk) are supposed to be independent. We consider these random sets as elements of the space Kd of all compact convex subsets of Rd with the Hausdorff distance. The boundary of Zα has non-trivial Cantor-type structure, and, in the case d=2 and under mild condition on σ, we prove that the Hausdorff dimension of the set of extremal points of the boundary of Zα is equal to α.As a by-product we provide some kind of geometrical characterization of (non-random) countable zonotopes in the class of zonoids, which are limits of zonotopes.

Arveson extreme points span free spectrahedra

Let $ SM_n(\mathbb{R})^g$ denote $g$-tuples of $n \times n$ real symmetric matrices. Given tuples $X=(X_1, \dots, X_g) \in SM_{n_1}(\mathbb{R})^g$ and $Y=(Y_1, \dots, Y_g) \in SM_{n_2}(\mathbb{R})^g$, a matrix convex combination of $X$ and $Y$ is a sum of the form \[ V_1^* XV_1+V_2^* Y V_2 \quad \quad \quad V_1^* V_1+V_2^* V_2=I_n \] where $V_1:\mathbb{R}^n \to \mathbb{R}^{n_1}$ and $V_2:\mathbb{R}^n \to \mathbb{R}^{n_2}$ are contractions. Matrix convex sets are sets which are closed under matrix convex combinations. A key feature of matrix convex combinations is that the $g$-tuples $X, Y$, and $V_1^* XV_1+V_2^* Y V_2$ do not need to have the same size. As a result, matrix convex sets are a dimension free analog of convex sets. While in the classical setting there is only one notion of an extreme point, there are three main notions of extreme points for matrix convex sets: ordinary, matrix, and absolute extreme points. Absolute extreme points are closely related to the classical Arveson boundary. A central goal in the theory of matrix convex sets is to determine if one of these types of extreme points for a matrix convex set minimally recovers the set through matrix convex combinations. This article shows that every real compact matrix convex set which is defined by a linear matrix inequality is the matrix convex hull of its absolute extreme points, and that the absolute extreme points are the minimal set with this property. Furthermore, we give an algorithm which expresses a tuple as a matrix convex combination of absolute extreme points with optimal bounds. Similar results hold when working over the field of complex numbers rather than the reals.

Existence of fixed points and retractions for asymptotically nonexpansive semigroups in locally convex spaces

In this paper, we will be concerned with a problem raised by Lau (Fixed Point Theory Appl, 2010) for the existence of common fixed points for a right reversible semitopological semigroup S acting asymptotically nonexpansive on a nonempty compact convex set in the framework of locally convex spaces. We give answers to this problem under the left amenability of \(\text {ARUC}(S)\) and \(\text {LMC}(S)\) and then study the existence of Q-nonexpansive retractions in the pointwise closure of S using a new method.

Circles and crossing planar compact convex sets

Let $K_0$ be a compact convex subset of the plane $\mathbb R^2$, and assume that whenever $K_1\subseteq \mathbb R^2$ is congruent to $K_0$, then $K_0$ and $K_1$ are not crossing in a natural sense due to L. Fejes-T\'oth. A theorem of L. Fejes-T\'oth from 1967 states that the assumption above holds for $K_0$ if and only if $K_0$ is a disk. In a paper appeared in 2017, the present author introduced a new concept of crossing, and proved that L. Fejes-T\'oth's theorem remains true if the old concept is replaced by the new one. Our purpose is to describe the hierarchy among several variants of the new concepts and the old concept of crossing. In particular, we prove that each variant of the new concept of crossing is more restrictive then the old one. Therefore, L. Fejes-T\'oth's theorem from 1967 becomes an immediate consequence of the 2017 characterization of circles but not conversely. Finally, a mini-survey shows that this purely geometric paper has precursor in combinatorics and, mainly, in lattice theory.

Intersection theorems for weak KKM set-valued mappings in the finite dimensional setting

Let X be a convex set in a vector space, Y be a nonempty set and S,T:X⇉Y two set-valued mappings. S is said to be a weak KKM mapping w.r.t. T if for each nonempty finite subset A of X and any x∈conv A, T(x)∩S(A)≠∅. Recently, the authors obtained two intersection theorems for a pair of such mappings, when X is a compact convex subset of a topological vector space. In this paper, we obtain open versions of the above mentioned theorems when X is a compact convex set in Rn. As applications, we establish several minimax inequalities and existence criteria for the solutions of three types of set-valued equilibrium problems.

Chord shortening flow and a theorem of Lusternik and Schnirelmann

We introduce a new geometric flow called the chord shortening flow which is the negative gradient flow for the length functional on the space of chords with end points lying on a fixed submanifold in Euclidean space. As an application, we give a simplified proof of a classical theorem of Lusternik and Schnirelmann (and a generalization by Riede and Hayashi) on the existence of multiple orthogonal geodesic chords. For a compact convex planar domain, we show that any convex chord which is not orthogonal to the boundary would shrink to a point in finite time under the flow.

Harmonic mappings valued in the Wasserstein space

We propose a definition of the Dirichlet energy (which is roughly speaking the integral of the square of the gradient) for mappings μ:Ω→(P(D),W2) defined over a subset Ω of Rp and valued in the space P(D) of probability measures on a compact convex subset D of Rq endowed with the quadratic Wasserstein distance. Our definition relies on a straightforward generalization of the Benamou-Brenier formula (already introduced by Brenier) but is also equivalent to the definition of Korevaar, Schoen and Jost as limit of approximate Dirichlet energies, and to the definition of Reshetnyak of Sobolev spaces valued in metric spaces.We study harmonic mappings, i.e. minimizers of the Dirichlet energy provided that the values on the boundary ∂Ω are fixed. The notion of constant-speed geodesics in the Wasserstein space is recovered by taking for Ω a segment of R. As the Wasserstein space (P(D),W2) is positively curved in the sense of Alexandrov we cannot apply the theory of Korevaar, Schoen and Jost and we use instead arguments based on optimal transport. We manage to get existence of harmonic mappings provided that the boundary values are Lipschitz on ∂Ω, uniqueness is an open question.If Ω is a segment of R, it is known that a curve valued in the Wasserstein space P(D) can be seen as a superposition of curves valued in D. We show that it is no longer the case in higher dimensions: a generic mapping Ω→P(D) cannot be represented as the superposition of mappings Ω→D.We are able to show the validity of a maximum principle: the composition F∘μ of a function F:P(D)→R convex along generalized geodesics and a harmonic mapping μ:Ω→P(D) is a subharmonic real-valued function.We also study the special case where we restrict ourselves to a given family of elliptically contoured distributions (a finite-dimensional and geodesically convex submanifold of (P(D),W2) which generalizes the case of Gaussian measures) and show that it boils down to harmonic mappings valued in the Riemannian manifold of symmetric matrices endowed with the distance coming from optimal transport.

Smooth convex extensions of convex functions

Let C be a compact convex subset of $$\mathbb {R}^n$$ , $$f:C\rightarrow \mathbb {R}$$ be a convex function, and $$m\in \{1, 2, \ldots , \infty \}$$ . Assume that, along with f, we are given a family of polynomials satisfying Whitney’s extension condition for $$C^m$$ , and thus that there exists $$F\in C^{m}(\mathbb {R}^n)$$ such that $$F=f$$ on C. It is natural to ask for further (necessary and sufficient) conditions on this family of polynomials which ensure that F can be taken to be convex as well. We give a satisfactory solution to this problem in the case $$m=\infty $$ , and make some remarks about the case of finite $$m\ge 2$$ . For a solution to a similar problem in the case $$m=1$$ (even for C not necessarily convex), see [1–3].

Discrete and continuous-time formulations for dealing with break periods: Preemptive and non-preemptive scheduling

This paper presents new mixed-integer linear programming (MILP) approaches for handling preemption both in discrete and continuous-time scheduling formulations. Preemption refers to the capability of interrupting the execution of a task when encountering a pre-defined break period, assuming that the task continues immediately after the end of such time window. We rely on Generalized Disjunctive Programming to derive the constraints for the continuous-time formulations and on a compact convex hull reformulation to make them computationally efficient. We investigate both the general precedence and multiple time grids representation concepts. Generalization of the discrete-time formulation is simpler, involving a change in the model parameters. Validation and comparison of the mathematical formulations is done through the solution of sixteen benchmark problems, involving instances with one to four sets of breaks. The results show that the general precedence formulation is computationally more effective for flexible flowshops, being outperformed by the discrete-time approach when considering common rather than machine-dependent breaks. For single stage plants with parallel units, the continuous multiple time grid formulation prevails.

A New gH-Difference for Multi-Dimensional Convex Sets and Convex Fuzzy Sets

In the setting of Minkowski set-valued operations, we study generalizations of the difference for (multidimensional) compact convex sets and for fuzzy sets on metric vector spaces, extending the Hukuhara difference. The proposed difference always exists and allows defining Pompeiu-Hausdorff distance for the space of compact convex sets in terms of a pseudo-norm, i.e., the magnitude of the difference set. A computational procedure for two dimensional sets is outlined and some examples of the new difference are given.