On Characterizations of $\sigma$-Quasiconvexity

TL;DR

This paper clarifies and completes the theoretical understanding of $\sigma$-quasiconvexity by providing corrected proofs and establishing full equivalence of first-order conditions, thus advancing the mathematical theory of quasiconvex functions.

Contribution

It offers corrected proofs of classical gradient characterizations and establishes the full equivalence of first-order conditions for differentiable $\sigma$-quasiconvexity, resolving open questions.

Findings

Corrected classical gradient characterizations of quasiconvexity.

Established full equivalence of first-order conditions for differentiable $\sigma$-quasiconvexity.

Provided a concise, self-contained proof of a classical characterization from the 1970s.

Abstract

We revisit classical gradient characterizations of quasiconvexity and provide corrected proofs that close gaps in earlier arguments. For the differentiable case of $\sigma$-quasiconvexity, we establish the full equivalence between several first-order conditions, resolving a remaining implication left open in the recent literature. Our approach yields a concise, self-contained proof of a classical characterization originally stated in the 1970s and sharpens the first-order theory for strong quasiconvexity.