Let $p_1,\ldots,p_n \in \mathbb{R}^n$ be linearly independent and $C=Conv\{p_1,-p_1,p_2,-p_2,\ldots,p_n,-p_n\}$. Is it true that C is combinatorially equivalent to the n-dimensional cross-polytope? (Because multiplication by a non-singular matrix shouldn't change the structure of the polytope.)
2026-03-29 10:08:00.1774778880
Conbinatorial equivalence to cross-polytope
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Yes, indeed. The argument you give seems sufficient to me. The change-of-basis transformation, being linear and invertible, induces a combinatorial isomorphism of $C$ with the standard cross-polytope.