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Let fn−1(P) denote the number of facets of a polytope P in R n. We show that there exist 0/1 polytopes P with fn−1(P) ≥ cn log 2 n n/2 where c > 0 is an absolute constant. This improves earlier work of Bárány and Pór on a question of Fukuda and Ziegler.
Let G be a semi-direct product G = A ×ϕ K with A Abelian and K compact. We characterize spread-out probability measures on G that are mixing by convolutions by means of their Fourier transforms. A key tool is a spectral radius formula for the Fourier transform of a regular Borel measure on G that we develop, and which is analogous to the well-known… (More)
We show that there exist 0/1 polytopes in R n whose number of facets exceeds cn log n n/2 , where c > 0 is an absolute constant.