Related papers: When is a Minkowski norm strictly sub-convex?

The Brunn-Minkowski and Pr\'{e}kopa-Leindler inequalities admit a variety of proofs that are inspired by convexity. Nevertheless, the former holds for compact sets and the latter for integrable functions so it seems that convexity has no…

We characterize all (absolute) 1-Lipschitz retracts Q of R^n with the maximum norm. Omitting two technical details, they coincide with the subsets written as the solution set of (at most) 2n inequalities like follows. For every coordinate…

Properties of two classes of generally convex sets in the n-dimentional real Euclidean space, called m-semiconvex and weakly m-semiconvex, 1<=m<n, are investigated in the present work. In particular, it is established that an open set with…

Let $\Omega $ be a bounded ${\mathcal{C}}^{\infty}$-smoothly bounded domain in ${\mathbb{C}}^{n}.$ For such a domain we define a new notion between strict pseudo-convexity and pseudo-convexity: the size of the set $W$ of weakly…

In this note we provide a simple proof of some properties enjoyed by convex functions having the engulfing property. In particular, making use only of results peculiar to convex analysis, we prove that differentiability and strict convexity…

Consider a convex polygon P in the plane, and denote by U a homothetical copy of the vector sum of P and (-P). Then the polygon U, as unit ball, induces a norm such that, with respect to this norm, P has constant Minkowskian width. We…

Considering smooth mappings from input vectors to continuous targets, our goal is to characterise subspaces of the input domain, which are invariant under such mappings. Thus, we want to characterise manifolds implicitly defined by level…

This note deals with certain properties of convex functions. We provide results on the convexity of the set of minima of these functions, the behaviour of their subgradient set under restriction, and optimization of these functions over an…

The real homology of a compact, n-dimensional Riemannian manifold M is naturally endowed with the stable norm. The stable norm of a homology class is the minimal Riemannian volume of its representatives. If M is orientable the stable norm…

In this paper, we prove an extended version of the Minkowski Inequality, holding for any smooth bounded set $\Omega \subset \mathbb R^n$, $n\geq 3$. Our proof relies on the discovery of effective monotonicity formulas holding along the…

We prove that, any problem of minimization of proper lower semicontinuous function defined on a normal Hausdorff space, is canonically equivalent to a problem of minimization of a proper weak * lower semicontinuous convex function defined…

We prove an abstract criterion that a surjective convolution operator in spaces of analytic functions on convex subsets of the complex plane has a continuous linear right inverse. Considered convex sets have a countable neighborhood basis…

Given an affine variety X and a finite dimensional vector space of regular functions L on X, we associate a convex body to (X, L) such that its volume is responsible for the number of solutions of a generic system of functions from L. This…

We prove that there exists an absolute constant $\alpha >1$ with the following property: if $K$ is a convex body in ${\mathbb R}^n$ whose center of mass is at the origin, then a random subset $X\subset K$ of cardinality ${\rm…

We prove that the Gauss map of a surface of constant mean curvature embedded in Minkowski space is harmonic. This fact will then be used to study 2+1 gravity for surfaces of genus higher than one. By considering the energy of the Gauss map,…

General results on convex bodies are reviewed and used to derive an exact closed-form parametric formula for the Minkowski sum boundary of $m$ arbitrary ellipsoids in $N$-dimensional Euclidean space. Expressions for the principal curvatures…

We introduce a new topological property called (*) and the corresponding class of topological spaces, which includes spaces with $G_\delta$-diagonals and Gruenhage spaces. Using (*), we characterise those Banach spaces which admit…

Each series $\sum_{n=1}^\infty a_n$ of real positive terms gives rise to a topology on $\mathbb{N} = \{1,2,3,...\}$ by declaring a proper subset $A\subseteq \mathbb{N}$ to be closed if $\sum_{n\in A} a_n < \infty$. We explore the…

Under study are some vector optimization problems over the space of Minkowski balls, i.e., symmetric convex compact subsets in Euclidean space. A typical problem requires to achieve the best result in the presence of conflicting goals;…

Let $E \subset \mathbb{R}^n$ be a compact set, and $f:E \to \mathbb{R}$. How can we tell if there exists a convex extension $F \in C^{1,1}(\mathbb{R}^n)$ of $f$, i.e. satisfying $F|_E = f|_E$? Assuming such an extension exists, how small…