Related papers: Characterization of self-polar convex functions
An $r^{\an}$-center of a compact body $\Om$ in an $n$ dimensional Euclidean space is a point that gives an extremal value of the regularized Riesz potential, which is the (Hadamard regularization of) integration on $\Om$ of the distance…
This paper considers projection and convolution operations for integrally convex functions, which constitute a fundamental function class in discrete convex analysis. It is shown that the class of integrally convex functions is stable under…
The notion of a valuation on convex bodies is very classical. The notion of a valuation on a class of functions was recently introduced and studied by M. Ludwig and others. We study an explicit relation between continuous valuations on…
We introduce a new integral invariant for isometric actions of compact Lie groups, the copolarity. Roughly speaking, it measures how far from being polar the action is. We generalize some results about polar actions in this context. In…
Suppose that u is the potential function of a complete K\"ahler-Einstein metric on a bounded strictly convex domain in $\mathbb{C}^n$. We prove that u itself is strictly convex.
We introduce notions of concavity for functions on balanced polyhedral spaces, and we show that concave functions on such spaces satisfy several strong continuity properties.
We prove a version of the Bernstein-Walsh theorem on uniform polynomial approximation of holomorphic functions on compact sets in several complex variables. Here we consider subclasses of the full polynomial space associated to a convex…
We prove two results on convex subsets of Euclidean spaces invariant under an orthogonal group action. First, we show that invariant spectrahedra admit an equivariant spectrahedral description, i.e., can be described by an equivariant…
Biunivalent holomorphic functions form an interesting class in geometric function theory and are connected with special functions and solutions of complex differential equations. The paper reveals a deep connection between biunivalence and…
We study the closed convex hull of various collections of Hilbert functions. Working over a standard graded polynomial ring with modules that are generated in degree zero, we describe the supporting hyperplanes and extreme rays for the…
Denote by $S^2$ the two-dimensional sphere. A spherical convex body on $S^2$ which does not properly contain a spherical convex body of the same spherical thickness is called a reduced body. We give three characterizations of reducedness of…
It is conjectured that all decomposable (i.e. interior can be triangulated without adding new vertices) polyhedra with vertices in convex position are infinitesimally rigid and only recently has it been shown that this is indeed true under…
We introduce the notion of Loewner (ellipsoid) function for a log concave function and show that it is an extension of the Loewner ellipsoid for convex bodies. We investigate its duality relation to the recently defined John (ellipsoid)…
We present the applications of variation -- wavelet analysis to polynomial/rational approximations for orbital motion in transverse plane for a single particle in a circular magnetic lattice in case when we take into account multipolar…
In the early 1960's, Moreau and Rockafellar introduced a concept of called \emph{subgradient} for convex functions, initiating the developments of theoretical and applied convex analysis. The needs of going beyond convexity motivated the…
In this paper we improve results related to Normalized Jensen Functional for convex functions and Uniformly Convex Functions.
Using the duality of positive cones, we show that applying the polar transform from convex analysis to local positivity invariants for divisors gives interesting and new local positivity invariants for curves. These new invariants have nice…
We experimentally study a monolayer of vibrated disks with a built-in polar asymmetry which enables them to move quasi-balistically on a large persistence length. Alignment occurs during collisions as a result of self-propulsion and hard…
In this paper, approximate convexity and approximate midconvexity properties, called $\varphi$-convexity and $\varphi$-midconvexity, of real valued function are investigated. Various characterizations of $\varphi$-convex and…
Given any polar pair of convex bodies we study its conjugate face maps and we characterize conjugate faces of non-exposed faces in terms of normal cones. The analysis is carried out using the positive hull operator which defines lattice…