On the volume of projections of the cross-polytope

2018/08/28 by Ivanov, G.
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.1808.09165

Abstract

We study properties of the volume of projections of the n-dimensional cross-polytope \crospⁿ = \ x ∈ \Rⁿ | |x₁| + … + |x_n| \leqslant 1\. We prove that the projection of \crospⁿ onto a k-dimensional coordinate subspace has the maximum possible volume for k=2 and for k=3. We obtain the exact lower bound on the volume of such a projection onto a two-dimensional plane. Also, we show that there exist local maxima which are not global ones for the volume of a projection of \crospⁿ onto a k-dimensional subspace for any n > k \geqslant 2.

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