On the volume of projections of the cross-polytope

Abstract

We study properties of the volume of projections of the $n$-dimensional cross-polytope $\crosp^n = \{ x \in \R^n \mid |x_1| + \dots + |x_n| \leqslant 1\}.$ We prove that the projection of $\crosp^n$ 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 $\crosp^n$ onto a $k$-dimensional subspace for any $n > k \geqslant 2.$