Related papers: Convex Sets and Minimal Sublinear Functions

Nearly convex sets play important roles in convex analysis, optimization and theory of monotone operators. We give a systematic study of nearly convex sets, and construct examples of subdifferentials of lower semicontinuous convex functions…

We look for minimizers of the buckling load problem with perimeter constraint in any dimension. In dimension 2, we show that the minimizing plates are convex; in higher dimension, by passing through a weaker formulation of the problem, we…

For a $n\times n$ unitary matrix $u=e^z$ with $z$ skew-Hermitian, the angles of $u$ are the arguments of its spectrum, i.e. the spectrum of $-iz$. For $1\le m\le n$, we show that $s_m(t)$, the sum of the first $m$ angles of the path…

Let $\Omega\subset\mathbb{R}^{n+1}$ have minimal Gaussian surface area among all sets satisfying $\Omega=-\Omega$ with fixed Gaussian volume. Let $A=A_{x}$ be the second fundamental form of $\partial\Omega$ at $x$, i.e. $A$ is the matrix of…

In this work we study subdivisions of $k$-rotationally symmetric planar convex bodies that minimize the maximum relative diameter functional. For some particular subdivisions called $k$-partitions, consisting of $k$ curves meeting in an…

Let $V$ be a finite nonempty set. A transit function is a map $R:V\times V\rightarrow 2^V$ such that $R(u,u)=\{u\}$, $R(u,v)=R(v,u)$ and $u\in R(u,v)$ hold for every $u,v\in V$. A set $K\subseteq V$ is $R$-convex if $R(u,v)\subset K$ for…

We prove that every closed characteristic of minimal action on the boundary of a uniformly convex domain in $\R^4$ bounds a disk-like global surface of section. A corollary is that the cylindrical symplectic capacity of a convex body in…

Let $G=(V(G),E(G))$ be a planar digraph embedded in the plane in which all inner faces are equilateral triangles (with three edges in each), and let the union $\Rscr$ of these faces forms a convex polygon. The question is: given a function…

We prove conditions for the existence of a continuous linear right inverse for a surjective convolution operator in spaces of germs of analytic functions on convex subsets of the complex plane. Considered convex sets have a countable…

Let $\Sigma$ be an axially symmetric, smooth, closed hypersurface in $\Bbb R^{n + 1}$ with a simply connected interior which is contained inside the unit sphere $\Bbb S^{n}$. For a continuous function $f$, which is defined on $\Bbb S^{n}$,…

Submodular function minimization is well studied, and existing algorithms solve it exactly or up to arbitrary accuracy. However, in many applications, such as structured sparse learning or batch Bayesian optimization, the objective function…

For a finite vector space $V$ and a non-negative integer $r\le\dim V$ we estimate the smallest possible size of a subset of $V$, containing a translate of every $r$-dimensional subspace. In particular, we show that if $K\subset V$ is the…

Given a compact set $K$ in the plane, which does not contain any triple of points forming a vertical and a horizontal segment, and a map $f\in C(K)$, we give a construction of functions $g,h\in C(\mathbb R)$ such that $f(x,y)=g(x)+h(y)$ for…

It is proved that the simplex is a strict local minimum for the volume product, P(K)=min(vol(K) vol(K^z)), K^z is the polar body of K with respect to z, the minimum is taken over z in the interior of K, in the Banach-Mazur space of…

Given an oriented Riemannian surface $(\Sigma, g)$, its tangent bundle $T\Sigma$ enjoys a natural pseudo-K\"{a}hler structure, that is the combination of a complex structure $\J$, a pseudo-metric $\G$ with neutral signature and a symplectic…

We prove that if a mapping F:X to Y, where X and Y are Banach spaces, is metrically regular at x for y and its inverse F^{-1} is convex and closed valued locally around (x,y), then for any function G:X to Y with lip G(x)regF(x|y)) < 1, the…

In [2], the authors prove Stillman's conjecture in all characteristics and all degrees by showing that, independent of the algebraically closed field $K$ or the number of variables, $n$ forms of degree at most $d$ in a polynomial ring $R$…

Given a set of line segments in the plane, not necessarily finite, what is a convex region of smallest area that contains a translate of each input segment? This question can be seen as a generalization of Kakeya's problem of finding a…

In this note we provide a full conjugacy and subdifferential calculus for convex convex-composite functions in finite-dimensional space. Our approach, based on infimal convolution and cone-convexity, is straightforward and yields the…

We give a proof of Ilmanen's lemma, which asserts that between a locally semi-convex and a locally semi-concave function it is possible to find a C$^{1,1}$ function.