A stability estimate for the Aleksandrov-Fenchel inequality, with an application to mean curvature
If \(K,L,C_ 1,...,C_{n-2}\) are convex bodies in \(E^ n\) (n\(\geq 3)\) and V denotes the mixed volume, the author introduces the nonnegative deficit \[ \Delta (K,L,C_ 1,...,C_{n-2})=V(K,L,C_ 1,...,C_{n- 2})^ 2-V(K,K,C_ 1,...,C_{n-2}) V(L,L,C_ 1,...,C_{n-2}) \] and denotes by \(\kappa^ n(r,R)\) the set of convex bodies in \(E^ n\) that contains some ball of radius r and are contained in some ball of radius R. Then, the following theorems are proved: 1. Let positive numbers \(\eta\),r,R and an integer \(p\in \{1,2,...,n-2\}\) be given. If \(K,L,C_ 1,...,C_ p\in \kappa^ n(r,R)\) and if \(C_ 1,...,C_ p\) are \(\eta\)-smooth, then \[ \Delta (K,L,C_ 1,...,C_ p,B,...,B)\geq c_ 1\eta^{4(2^ p-1)}\Delta (K,L,B,...,B)^{2p} \] where the constant \(c_ 1\) depends only on n, p, r, R. 2. Let \(K\subset E^ n\) be a convex body contained in some ball of radius R, let \(0<\epsilon_ 0<1\). Suppose that the boundary of K is twice continuously differentiable and that the mean curvature H satisfies \(1-\epsilon \leq H\leq 1+\epsilon,\) for some positive \(\epsilon \leq \epsilon_ 0\). Then, there is a ball \(B_ 1\) of radius 1 such that \(\delta (K,B_ 1)\leq c_ 2\epsilon^ q\) with \(q=[(n+3)2^{n-3}]^{- 1},\) where \(c_ 2\) is a constant depending only on n, R, \(\epsilon_ 0\) and \(\delta\) denotes Hausdorff metric.
scientific article; zbMATH DE number 4173788
scientific article