Relative entropy of cone measures and $L_p$ centroid bodies

Details Relative entropy of cone measures and $L_p$ centroid bodies Relative entropy of cone measures and $L_p$ centroid bodies

2009-09-24

arxiv.org/abs/0909.4361v1

Mathematics

Functional Analysis

Scientific paper

Let $K$ be a convex body in $\mathbb R^n$. We introduce a new affine invariant, which we call $\Omega_K$, that can be found in three different ways: as a limit of normalized $L_p$-affine surface areas, as the relative entropy of the cone measure of $K$ and the cone measure of $K^\circ$, as the limit of the volume difference of $K$ and $L_p$-centroid bodies. We investigate properties of $\Omega_K$ and of related new invariant quantities. In particular, we show new affine isoperimetric inequalities and we show a "information inequality" for convex bodies.

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