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

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

2008-09-23

arxiv.org/abs/0809.3921v1

Mathematics

Functional Analysis

Scientific paper

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}.

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