Counterexample to a conjecture of Ziegler concerning a simple polytope and its dual

Abstract: Problem 4.19 in Ziegler's "Lectures on Polytopes" asserts that every simple $3$-dimensional polytope has the property that its dual can be constructed as the convex hull of a subset of the vertices of the original simple polytope. In this note we state a higher-dimensional analogue of this conjecture and provide a family of counterexamples for dimension $d \geq 3$.