The \(L_p\) affine surface areas are central notions of the \(L_p\) Brunn–Minkowski theory. They involve combinations of the cone measures of a convex body and its polar. We discuss their role in convex and differential geometry and in applications, their affine isoperimetric inequalities and their relation to information theory. Related quantities are the maximal, respectively minimal, inner and outer affine surface areas. Those share all the good properties of \(L_p\) affine surface area, but in addition have better continuity behavior. In dimension 2 and for \(p=1\), the maximal affine surface area was investigated by Baranyi, who, in this setting, also found a connection to the limit shape of lattice polytopes. In higher dimensions no such results are known. We use a thin shell estimate of Guedon and E. Milman to give asymptotic estimates on the size of the extremal affine surface areas.
Please email Anton Bernshteyn (abernsht ~at~ math.cmu.edu) for a password.
The ACO Seminar (2019–2020)