Exterior convexity and classical calculus of variations

TL;DR

This paper explores the connections between various notions of exterior convexity and classical convexity concepts in calculus of variations, introducing a new projection map to analyze their relationships.

Contribution

It introduces a generalized projection map and uses it to relate exterior convexity notions with classical convexity concepts, providing new proofs of existing results.

Findings

Established relations between exterior convexity and classical convexity notions.

Introduced a new projection map with algebraic properties.

Provided simplified proofs for previous results.

Abstract

We study the relation between various notions of exterior convexity introduced in Bandyopadhyay-Dacorogna-Sil \cite{BDS1} with the classical notions of rank one convexity, quasiconvexity and polyconvexity. To this end, we introduce a projection map, which generalizes the alternating projection for two-tensors in a new way and study the algebraic properties of this map. We conclude with a few simple consequences of this relation which yields new proofs for some of the results discussed in Bandyopadhyay-Dacorogna-Sil \cite{BDS1}.