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New Characterizations of Strong Convexity

Chadi Nour, Jean Takche

TL;DR

This work characterizes $r$-strong convexity for nonempty closed sets $A\subset\mathbb{R}^n$ by showing that $A$ is $r$-strongly convex iff it is $r$-spherically supported with $\,\operatorname{int} A\neq\emptyset$, and derives corollaries stating that epi-Lipschitzness (or convexity) suffices for the spherical-support equivalence. It also offers a second characterization: $A$ is $r$-strongly convex iff $A$ is convex, bounded, and $r$-negatively $\mathcal{E}_r(A)$-convex with $\mathcal{E}_r(A)=\{x:\,\text{dfar}_A(x)>r\}$. The proofs combine nonsmooth analysis with convex-geometric arguments, including a componentwise decomposition and analysis of proximal normals and farthest-point geometry. Overall, the results extend regularity concepts like prox-regularity and the exterior sphere condition to precise geometric criteria for strong convexity, with potential implications for control-theoretic reachable sets and related optimization problems.

Abstract

Parallel to the main results of [13] and [14], which explore the equivalence between prox-regularity, the exterior sphere condition, and $S$-convexity, we present novel characterizations of the $r$-strong convexity property, namely, of the sets that can be expressed as the intersection of closed balls with the same radius $r>0$.

New Characterizations of Strong Convexity

TL;DR

This work characterizes -strong convexity for nonempty closed sets by showing that is -strongly convex iff it is -spherically supported with , and derives corollaries stating that epi-Lipschitzness (or convexity) suffices for the spherical-support equivalence. It also offers a second characterization: is -strongly convex iff is convex, bounded, and -negatively -convex with . The proofs combine nonsmooth analysis with convex-geometric arguments, including a componentwise decomposition and analysis of proximal normals and farthest-point geometry. Overall, the results extend regularity concepts like prox-regularity and the exterior sphere condition to precise geometric criteria for strong convexity, with potential implications for control-theoretic reachable sets and related optimization problems.

Abstract

Parallel to the main results of [13] and [14], which explore the equivalence between prox-regularity, the exterior sphere condition, and -convexity, we present novel characterizations of the -strong convexity property, namely, of the sets that can be expressed as the intersection of closed balls with the same radius .
Paper Structure (3 sections, 12 theorems, 53 equations, 2 figures)

This paper contains 3 sections, 12 theorems, 53 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proofs of the main results

Key Result

Theorem 1.1

Let $A\subset\mathbb{R}^n$ be a nonempty and closed set not reduced to a singleton, and let $r>0$. Then $A$ is $r$-strongly convex if and only if $A$ is $r$-spherically supported with $\textnormal{int}\, A\not=\emptyset$.

Figures (2)

  • Figure 1: Proof of Lemma \ref{['epiofA']}
  • Figure 2: Proof of Lemma \ref{['lem3']}

Theorems & Definitions (13)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Lemma 2.1: clsw
  • Lemma 2.2: JCA2009
  • Proposition 2.3
  • Remark 2.1
  • Proposition 2.4: schneider
  • Lemma 3.1
  • ...and 3 more