A remark on the structure of the Busemann representative of a polyconvex function
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 R5 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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