A simple proof of an approximation theorem of H. Minkowski
A theorem of H. Minkowski states that every convex body in \(\mathbb{R}^ n\) can be approximated by a sequence of convex bodies whose boundaries are analytic \((n-1)\)-dimensional manifolds. The author gives a simple proof of the following stronger theorem. Theorem 1.1. Let \(C\) be a convex body in \(\mathbb{R}^ n\) with polar \(C^*\). Then there exists a sequence \(C_ k\) of convex bodies with the following properties: (1) \(C_ k\downarrow C\), (2) \(\partial C_ k\) and \(\partial C_ k^*\) are analytic \((n-1)\)- dimensional manifolds, (3) The curvatures of \(\partial c_ k\) and \(\partial C_ k^*\) never vanish. He also proves some further properties of \(C_ k\).
scientific article; zbMATH DE number 493319