Convex hypersurfaces with bounded first mean curvature measure (Q1291794)

Some \(C^\infty\)-approximation results for convex sets with bounded first mean curvature density are obtained and a strong maximum principle for some convex hypersurfaces is derived. If \(K\subset\mathbb{R}^n\) is a compact convex set whose boundary has first mean curvature density bounded below, then a concave function is constructed on \(K\) such that \(u/\partial K=0\) and \(\text{graph} (u)\) has first mean curvature density bounded below. The main theorem states that for any noncompact closed convex set \(C\subset \mathbb{R}^{n+1}\), \(n\geq 2\), with \(C^2\)-boundary whose first mean curvature satisfies a pinching condition, there exist \(2\leq k\leq n\) and a compact set \(K\subset \mathbb{R}^k\) such that \(C\) is congruent to \(K\times \mathbb{R}^{n+1-k}\).