Related papers: Functions with strictly convex epigraph
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…
This paper presents a general study of one-dimensional differentiability for functionals defined on convex domains that are not necessarily open. The local approximation is carried out using affine functionals, as opposed to linear…
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 show that domains, that allow for convex functions with unbounded gradient at their boundary, are convex.
We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid, and simplex. Each of…
These informal notes are concerned with spaces of functions in various situations, including continuous functions on topological spaces, holomorphic functions of one or more complex variables, and so on.
Discrete convex functions are used in many areas, including operations research, discrete-event systems, game theory, and economics. The objective of this paper is to offer a survey on fundamental operations for various kinds of discrete…
We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological…
In this paper, strongly $(\alpha,T)$-convex functions, i.e., functions $f:D\to \R$ satisfying the functional inequality $$ f(tx+(1-t)y)\leq tf(x)+(1-t)f(y)-t\alpha\big((1-t)(x-y)\big)-(1-t)\alpha\big(t(y-x)\big)$$ for $x,y\in D$ and $t\in…
This study focuses on Concave mappings, a class of univalent functions that exhibit a unique property: they map the unit disk onto a domain whose complement is convex. The main objective of this work is to characterize these mappings in…
An example is given of a hyperconvex manifold without non-constant bounded holomorphic functions, which is realized as a domain with real-analytic Levi-flat boundary in a projective surface.
Let $\Delta_m$ be the standard $m$-dimensional simplex of non-negative $m+1$ tuples that sum to unity and let $S$ be a nonempty subset of $\Delta_m$. A real valued function $h$ defined on a convex subset of a real vector space is $S$-almost…
In this paper, we prove a rigidity theorem for smooth strictly convex domains in Euclidean spaces.
The aim of this paper is to present an original approach that takes advantage from the geometric features of strictly convex functions to tackle the problem of finding the minimum from another perspective. The general idea is that near the…
In this short note we show that the tetrablock is i $\C$-convex domain. In the proof of this fact a new class of ($\C$-convex) domains is studied. The domains are natural caniddates to study on them the behavior of holomorphically invariant…
The aim of this paper is twofold. Firstly, we give easy-to-handle criteria to determine whether a given family of subsets of a vector space is a neighbourhood basis of the origin for a complete vector topology. Then, we apply these criteria…
Comparison and localization results for the Lempert function, the Carath\'eodory distance and their infinitesimal forms on strongly pseudoconvex domains are obtained. Related results for visible and strongly complete domains are proved.
We consider the space of convex functions defined in the Euclidean $n$-dimensional space, which are lower semi-continuous and tend to infinity at infinity. We study real-valued valuations defined on this space of functions, which are…
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…
Integrally convex functions constitute a fundamental function class in discrete convex analysis, including M-convex functions, L-convex functions, and many others. This paper aims at a rather comprehensive survey of recent results on…