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In a later paper Adiprasito and Izmestiev prove the following stronger result: "a triangulation of a sphere or ball is PL if and only if it becomes polytopal after sufficently many derived subdivisions." arxiv.org/abs/1311.2965
Any triangulation (or subdivision) defined by hyperplane cuts, such as the one in the answer, is regular. The sum of distances to the hyperplanes is a valid height function.
