Contraction of convex hypersurfaces by their affine normal
The author considers an affine invariant evolution equation for convex hypersurfaces in Euclidean space by assigning to each point a velocity equal to the unimodular 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. Moreover, it is also shown that the hypersurfaces converge smoothly to an ellipsoid after rescaling about the final point so to make the enclosed volume constant. Finally, the author uses this result in order to give new proofs of two known affine geometric consequences: the Blaschke-Santaló-Deicke affine isoperimetric inequality and the so-called Blaschke-Santaló inequality.
scientific article; zbMATH DE number 895024