An intrinsic volume metric for the class of convex bodies in ℝn

Abstract

A new intrinsic volume metric is introduced for the class of convex bodies in [Formula: see text]. As an application, an inequality is proved for the asymptotic best approximation of the Euclidean unit ball by arbitrarily positioned polytopes with a restricted number of vertices under this metric. This result improves the best known estimate, and shows that dropping the restriction that the polytope is contained in the ball or vice versa improves the estimate by at least a factor of dimension. The same phenomenon has already been observed in the special cases of volume, surface area and mean width approximation of the ball.

Asymptotic expected T$T$‐functionals of random polytopes with applications to Lp$L_p$ surface areas

“…Interestingly, in high dimensions the random approximation of smooth convex bodies is asymptotically as good as the best approximation as $$N\rightarrow \infty$$, up to absolute constants; see [1, 3–6, 9, 10, 16, 17, 20, 21, 24]. Choosing the minimizing density $$f=f_{\rm unif}$$ in Corollary 1.4, by Stirling's inequality we derive $$$$\begin{align*} \limsup _{N\rightarrow \infty }N^{\frac{2}{n-1}}\min _{Q\in \mathcal {P}_{n,N}^{\rm in}}\Delta _{S_p}(B_n,Q) &\le \limsup _{N\rightarrow \infty }N^{\frac{2}{n-1}}\mathbb {E}[\Delta _{S_p}(B_n,Q_{n,N}^{f_{\rm unif}})]\\ &\le \frac{1}{2}(n-p)\mu _{\partial B_n}(\partial B_n){\left(1+O{\left(\frac{\ln n}{n}\right)}\right)}.$$…”

On Minkowski and Blaschke symmetrizations of functions and related applications

“…The mean width metric has applications to the approximation of convex bodies by polytopes; we refer the reader to [5,6,35,48] and the references therein for some examples. Given K ∈ K n and an integer N ≥ n + 1, let P N (K) denote the set of all polytopes contained in K with at most N vertices.…”