Related papers: Dido's Problem and Beyond
We treat the classical notion of convexity in the context of hard real analysis. Definitions of the concept are given in terms of defining functions and quadratic forms, and characterizations are provided of different concrete notions of…
The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved…
The topics of Convexity and Concavity and Envelopes are central in Complex Analysis and extensively investigated. The aim of this paper is to find a possible counterpart in Algebraic Geometry. The article presents preliminary results on…
The work consists of solutions of metric problems for convex and finite subsets of geodesic spaces.
Some conjectures and open problems in convex geometry are presented, and their physical origin, meaning, and importance, for quantum theory and generic statistical theories, are briefly discussed.
Convexity is an important notion in non linear optimization theory as well as in infinite dimensional functional analysis. As will be seen below, very simple and powerful tools will be derived from elementary duality arguments (which are…
The paper studies possible functional analogs of classical problems from convex geometry. In particular, we provide some bounds in the functional Shephard, Busemann-Petty, and Milman problems generalizing known bounds in this problems for…
Under study is the new class of geometrical extremal problems in which it is required to achieve the best result in the presence of conflicting goals; e.g., given the surface area of a convex body $\mathfrak x$, we try to maximize the…
We investigate metric projections and distance functions referring to convex bodies in finite-dimensional normed spaces. For this purpose we identify the vector space with its dual space by using, instead of the usual identification via the…
We consider scalar equilibrium problems governed by a bifunction in a finite-dimensional framework. By using classical arguments in Convex Analysis, we show that under suitable generalized convexity assumptions imposed on the bifunction,…
The theory of abstract convexity, also known as convexity without linearity, is an extension of the classical convex analysis. There are a number of remarkable results, mostly concerning duality, and some numerical methods, however, this…
We consider the problem of approximating the solution of variational problems subject to the constraint that the admissible functions must be convex. This problem is at the interface between convex analysis, convex optimization, variational…
This paper deals with a variation of the classical isoperimetric problem in dimension $N\ge 2$ for a two-phase piecewise constant density whose discontinuity interface is a given hyperplane. We introduce a weighted perimeter functional with…
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.…
Some features of extended loops are considered. In particular, the behaviour under diffeomorphism transformations of the wavefunctions with support on the extended loop space are studied. The basis of a method to obtain analytical…
A review of author's work on cyclic inequalities of Shapiro-Diananda type and related optimization problems is presented.
Generalized polyhedral convex optimization problems in locally convex Hausdorff topological vector spaces are studied systematically in this paper. We establish solution existence theorems, necessary and sufficient optimality conditions,…
In this paper we develop a geometric approach to convex subdifferential calculus in finite dimensions with employing some ideas of modern variational analysis. This approach allows us to obtain natural and rather easy proofs of basic…
We give a new characterization of pseudoconvex point, and of finite type point, using analytic discs.
This is a survey on algorithmic questions about combinatorial and geometric properties of convex polytopes. We give a list of 35 problems; for each the current state of knowledege on its theoretical complexity status is reported. The…