ABOUT THE MEAN WIDTH OF SIMPLICES

We are interested in the maximal mean width of simplices in Ed having edge-length at most one. Probably the maximum is provided by the regular simplex with edge-length one. We prove it for d ≤ 5 and support this conjecture with some additional arguments

Counting the Faces of Randomly-Projected Hypercubes and Orthants, with Applications

Abstract. Let A be an n by N real-valued matrix with n < N; we count the number of k-faces fk(AQ) when Q is either the standard N-dimensional hypercube IN or else the positive orthant RN +. To state results simply, consider a proportional-growth asymptotic, where for fixed δ, ρ in (0, 1), we have a sequence of matrices An,Nn and of integers kn with n/Nn → δ, kn/n → ρ as n → ∞. If each matrix An,Nn has its columns in general position, then fk(AIN)/fk(I N) tends to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). Also, if each An,Nn is a random draw from a distribution which is invariant under right multiplication by signed permutations, then fk(ARN +)/fk(RN +) tends almost surely to zero or one depending on whether ρ> min(0, 2 − δ−1) or ρ < min(0, 2 − δ−1). We make a variety of contrasts to related work on projections of the simplex and/or cross-polytope. These geometric face-counting results have implications for signal processing, information theory, inverse problems, and optimization. Indeed, face counting is related to conditions for uniqueness of solutions of underdetermine

Asymptotic formulae for the lattice point enumerator

Let M be a convex body such that the boundary has positive curvature. Then by a well developed theory dating back to Landau and Hlawka for large the number of lattice points in M is given by G(M) = V (M) + O( d\Gamma1\Gamma"(d)) for some positive "(d). Here we give for general convex bodies the weaker estimate jG(M) \Gamma V (M)