  Given a convex body K in R^n and p in R, we introduce and study the extremal
inner and outer affine surface areas
  IS_p(K) = sup_{K'\subseteq K} (as_p(K') ) and os_p(K)=inf_{K'\supseteq K}
(as_p(K') ), 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
Barany. 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 of Guedon and Milman and the L\"owner ellipsoid
to give asymptotic estimates on the size of IS_p(K) and os_p(K). Surprisingly,
both quantities are proportional to a power of volume.
