The isoperimetric profile of a convex body assigns to each volume the least perimeter, measured inside the body's interior, of a subset of that volume. Students learn how this profile can be controlled without assuming the boundary is smooth: Hausdorff convergence of convex bodies is upgraded to Lipschitz convergence by explicit bilipschitz radial maps whose dilatation depends only on the circumradius-to-inradius ratio, which yields continuity and concavity of the profile, connectedness of isoperimetric regions, Hausdorff convergence of those regions and their free boundaries, and the small-volume asymptotics in which the profile approaches that of the tangent cone of least solid angle.
A convex body is a compact convex set in Euclidean space with non-empty interior, and its isoperimetric profile assigns to each volume v the smallest perimeter, measured only inside the interior of t…