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This article exhibits a 4-dimensional combinatorial polytope that has noantiprism, answering a question posed by Bernt Lindst\"om. As a consequence,any realization of this combinatorial polytope has a face that it cannot restupon without toppling over. To this end, we provide a general method forsolving a broad class of realizability problems. Specifically, we show that forany semialgebraic property that faces inherit, the given property holds forsome realization of every combinatorial polytope if and only if the propertyholds from some projective copy of every polytope. The proof uses the followingresult by Below. Given any polytope with vertices having algebraic coordinates,there is a combinatorial "stamp" polytope with a specified face that isprojectively equivalent to the given polytope in all realizations. Here weconstruct a new stamp polytope that is closely related to Richter-Gebert'sproof of universality for 4-dimensional polytopes, and we generalize severaltools from that proof.

Antiprismless, or: Reducing Combinatorial Equivalence to Projective Equivalence in Realizability Problems for Polytopes