For each convex body and parameter p in the meaningful range, the largest or smallest p-affine surface area among bodies inside or outside it grows like a fixed power of its volume, up to dimension-dependent constants.
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Constrained convex bodies with extremal affine surface areas
For each convex body and parameter p in the meaningful range, the largest or smallest p-affine surface area among bodies inside or outside it grows like a fixed power of its volume, up to dimension-dependent constants.