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"... There exist positive constants c0 and c1 = c1(n) such that for every 0 < ɛ < 1/2 the following holds: Let P be a convex polytope in R n having N ≥ cn 0 /ɛ vertices x1,..., xN. Then there exists a subset A ⊂ {1,..., N}, card (A) ≥ (1 − 2ɛ)N, such that for all i ∈ A voln(P) − voln(conv(vert (P) \ { ..."
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There exist positive constants c0 and c1 = c1(n) such that for every 0 < ɛ < 1/2 the following holds: Let P be a convex polytope in R n having N ≥ cn 0 /ɛ vertices x1,..., xN. Then there exists a subset A ⊂ {1,..., N}, card (A) ≥ (1 − 2ɛ)N, such that for all i ∈ A voln(P) − voln(conv(vert (P) \ {xi})) voln(P) n+1 n+1 ≤ c1(n)ɛ n−1 N n−1. Also, if P is a convex polytope in R n having N ≥ cn 0 /ɛ facets. Let H+ i be the half space determined by the facet Fi, which contains P (i = 1,..., N). Then there exists a subset A ⊂ {1,..., N}, card (A) ≥ (1 − 2ɛ)N, such that for all i ∈ A voln 1
