A remark on the structure of the Busemann representative of a polyconvex function

TL;DR

This paper investigates the structure of the Busemann representative of a polyconvex function on 2x2 matrices, revealing that it is generally strictly larger than the natural convex representative, which has implications for convex analysis.

Contribution

It proves that the Busemann representative of a polyconvex function can be strictly larger than the natural convex representative, clarifying its structure.

Findings

The Busemann representative f is generally strictly larger than the natural convex representative g.

f can be expressed as a supremum over affine functions bounded by W.

The result highlights a surprising difference in convex representatives of polyconvex functions.

Abstract

Let W be a polyconvex function defined on the 2 x 2 real matrices. The Busemann representative f, say, of W is the largest possible convex representative of W. Writing L for the set of affine functions on R^{5} such that a(A, det A) is less than or equal to W(A) for all 2 x 2 real matrices A, f can then be expressed as f(X) = sup {a(X): a lies in L}. This short note proves the surprising result that f is in general strictly larger than the `natural' convex representative g(X) = sup {a(X): a lies in L and a(A, det A)=W(A) for some A}.