In Euclidean space \({\mathbb R}^d\), let \(P\) be a convex polytope containing the origin in its interior, and let \(P^\circ\) denote the polar polytope of \(P\). The polytope \(P\) is self-dual if it is combinatorially isomorphic to \(P^\circ\). In this paper, a subset of the self-dual polytopes is defined. A polytope \(P\) is called self-polar if \(P= UP^\circ\) for some orthogonal linear transformation \(U\). The article is devoted to the study of self-polar polytopes, with a special view to negatively self-polar polytopes (i.e., satisfying \(P=-P^\circ\)). The two-dimensional case is settled in an elementary way, together with the determination of all corresponding maps \(U\). In three dimensions, somewhat surprisingly, it is shown (based on the Koebe-Andreev-Thurston theorem), that every self-dual polytope has a self-polar realization (the higher-dimensional analogue is later posed as an open question). In higher dimensions, several constructions of self-polar polytopes are described. Also ways to build lower-dimensional self-polar polytopes from higher-dimensional ones are considered. Further, it is shown that for \(d\ge 3\) there exist negatively self-polar \(d\)-polytopes with \(n\) vertices for all values of \(n\ge d+1\) except \(n=d+2\). Of the outcomes of some more complicated constructions, we mention only the following result. For a polytope \(P\subset{\mathbb R}^d\) such that \(P\subseteq -P^\circ\), there exists a polytope \(Q\) such that \(P\subseteq Q=-Q^\circ\subseteq-P^\circ\). The paper concludes with some open questions.
For the entire collection see [Zbl 1467.52001].