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Description

Given a convex body $K$ in $\mathbb R^n$ and a real number $p$, we study the extremal inner and outer affine surface areas $$IS_p(K) = \sup_{K'\subseteq K}\big(as_p(K')\big)$$ and $$os_p(K)=\inf_{K'\supseteq K}\big(as_p(K')\big),$$ where $as_p(K')$ denotes the $L_p$-affine surface area of $K'$, and the supremum is taken over all convex subsets of $K$ and the infimum over all convex compact subsets containing $K$. The convex body that realizes $IS_1(K)$ in dimension 2 was determined by B\'ar\'any. He also showed that this body is the limit shape of lattice polytopes in $K$. In higher dimensions no results are known about the extremal bodies. We use a thin shell estimate to give asymptotic estimates on the size of $IS_p(K)$. We use the L\"owner ellipsoid of $K$ to give asymptotic estimates on the size of $os_p(K)$. Based on joint work with Ohad Giladi, Han Huang and Carsten Schuett.