Error of asymptotic formulae for volume approximation of convex bodies in \({\mathbb E}^d\)
In the paper under review the error of asymptotic formulae for volume approximation of a sufficiently smooth convex body \(C\subset {\mathbb R}^d\) by polytopes \(P_n\) is estimated as the number \(n\) of facets of \(P_n\) tends to infinity. The author [Forum Math. 5, No. 6, 521-537 (1993; Zbl 0788.41020)] derived a precise asymptotic formula for the volume of the symmetric difference, \(\delta(C,P_n)\), of \(C\) and its best approximating polytopes \(P_n\). The order of approximation is \(\displaystyle a(C)/ n^{2/(d-1)}\) where \(a(C)>0\) depends on the affine surface area of \(C\). Here the author refines his result and shows that the error term \[ \delta(C,P_n)-a(C)/n^{2/(d-1)}=O(\displaystyle 1/ n^{2/(d-1) + 1/(3 (d-1))-\varepsilon}) \] as \(n\to\infty\) for every \(\varepsilon>0\). He conjectures that the precise order of this error term is \(a'(C)/n^{3/(d-1)}\). The result is applied to estimate the minimum isoperimetric quotient of polytopes \(P_n\) in Minkowski geometry as \(n\to\infty\).
scientific article; zbMATH DE number 1773410