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Two characterizations of quasiconvexity

Włodzimierz Fechner

TL;DR

The work addresses characterizations of quasiconvexity under minimal regularity for radially semicontinuous mappings on convex subsets of real linear spaces. It develops two complementary characterizations: a Daróczy–Palés type structural description of the non-quasiconvex set for radially lower semicontinuous functions using the set $T(x,y)$ and its decomposition, and a local-max based criterion for radially upper semicontinuous functions via the absence of certain strict local maxima. These yield an extension of Sion's minimax theorem under segment-based inequalities and two new characterizations of quasiconvex risk measures grounded in radial semicontinuity and diversification/local-max properties. The results broaden the applicability of quasiconvexity in optimization and risk assessment by relaxing regularity assumptions and linking theoretical conditions to practical concepts like diversification and local risk maximization.

Abstract

We present two characterizations of quasiconvexity for radially semicontinuous mappings defined on a convex subset of a real linear space. As an application we obtain an extension of the Sion's minimax theorem, as well as a new characterization of quasiconvex risk measures.

Two characterizations of quasiconvexity

TL;DR

The work addresses characterizations of quasiconvexity under minimal regularity for radially semicontinuous mappings on convex subsets of real linear spaces. It develops two complementary characterizations: a Daróczy–Palés type structural description of the non-quasiconvex set for radially lower semicontinuous functions using the set and its decomposition, and a local-max based criterion for radially upper semicontinuous functions via the absence of certain strict local maxima. These yield an extension of Sion's minimax theorem under segment-based inequalities and two new characterizations of quasiconvex risk measures grounded in radial semicontinuity and diversification/local-max properties. The results broaden the applicability of quasiconvexity in optimization and risk assessment by relaxing regularity assumptions and linking theoretical conditions to practical concepts like diversification and local risk maximization.

Abstract

We present two characterizations of quasiconvexity for radially semicontinuous mappings defined on a convex subset of a real linear space. As an application we obtain an extension of the Sion's minimax theorem, as well as a new characterization of quasiconvex risk measures.
Paper Structure (6 sections, 10 theorems, 23 equations)

This paper contains 6 sections, 10 theorems, 23 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Applications
  4. Sion's minimax theorem
  5. Quasiconvex risk measures
  6. Conclusions

Key Result

Theorem 1

Suppose that $f\colon D \to \overline{\mathbb{R}}$ is a radially lower semicontinuous function. Then for every $x, y \in D$ there exists an at most countable set $A=A(x,y)$ (possibly empty) and the points $u_i, v_i \in [x,y], i \in A$ such that:

Theorems & Definitions (20)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Corollary 2
  • Remark 1
  • Example 1
  • Example 2
  • Theorem 2
  • proof
  • ...and 10 more