Contraction of convex hypersurfaces by their affine normal

Abstract

An affine-invariant evolution equation for convex hypersurfaces in Euclidean space is defined by assigning to each point a velocity equal to the affine normal vector. For an arbitrary compact, smooth, strictly convex initial hypersurface, it is shown that this deformation produces a unique, smooth family of convex hypersurfaces, which converge to a point in finite time. Furthermore, the hypersurfaces converge smoothly to an ellipsoid after rescaling about the final point to make the enclosed volume constant. The result leads to simple proofs of some affine-geometric isoperimetric inequalities. Show more