Dropping a vertex or a facet from a convex polytope (Q2716422)

As it is well known any convex body \(C\) in \(\mathbb{R}^n\) can be approximated by a polytope \(P\subset C\) with at most \(N\) vertices so that NEWLINE\[NEWLINE{vol_n (C)-vol_n(P) \over vol_n(C)}\leq {cn\over N^{2/(n-1)}},NEWLINE\]NEWLINE where \(vol_n (\cdot)\) denotes the \(n\)-dimensional volume and \(c\) a suitable constant. An analogous result holds for the outer approximating polytope with at most \(N\) facets. This article concentrates on the case where \(C\) is itself a polytope with \(N\) vertices and \(M\) facets and \(P\) is a polytope with fewer vertices or facets, with a particular effort to control the size of the constants involved in the estimates.NEWLINENEWLINENEWLINEThe main results are as follows. There exist positive constants \(c_0\) and \(c_1=c_1(n)\) such that for every \(0<\varepsilon <1/2\) the following holds: Let \(P\) be a convex polytope in \(\mathbb{R}^n\) having \(N\geq c^n_0/ \varepsilon\) vertices. Then there exists a subset \(A\subset \{1,\dots,N\}\), \(\text{card} (A)\geq(1-2 \varepsilon)N\), such that for all \(i\in A\) NEWLINE\[NEWLINE{vol_n(P)- vol_n(Q) \over vol_n(P)} \leq c_1(n)\varepsilon^{- (n+1)/(n-1)} N^{-(n+1)/(n-1)},NEWLINE\]NEWLINE where \(Q\) is the convex hull of all the vertices of \(P\) other than \(x_i\).NEWLINENEWLINENEWLINEAlso, if \(P\) is a convex polytope having \(N\geq c_0^n/ \varepsilon\) facets and \(H^+_i\) is the half space determined by the facet \(F_i\) \((i=1, \dots,N)\) and containint \(P\), then there exists a subset \(A\subset\{1, \dots,N\}\), \(\text{card} (A)\geq(1-2 \varepsilon)N\), such that for all \(i\in A\) NEWLINE\[NEWLINE{vol_n (\bigcap_{j\neq i} H_j^+)-vol_n (P)\over vol_n (P)}\leq c_1(n) \varepsilon^{-(n+1)/(n-1)} N^{-(n+1)/(n-1)}.NEWLINE\]NEWLINE The two dimensional case has been previously studied in [\textit{M. A. Lopez} and \textit{S. Reisner}, Int. J. Comput. Geom. Appl. 10, No. 5, 445-452 (2000; Zbl 0973.65013)].NEWLINENEWLINENEWLINEAs a successive application there is a method to construct inner or outer approximating polytopes with fewer vertices or facets than the given polytope.