Let's say I have a combinatorially self-dual polytope $P\subseteq\Bbb R^d$, i.e., its face lattice is isomorphic to its dual (you reverse the direction of the lattice order).

Question: Is it always possible to realize $P$ geometrically, so that $P$ and its polar $$P^\circ := \{x\in\Bbb R^d \mid \langle x,s\rangle \le 1 \text{ for all $s\in P$}\}$$ are geometrically related in some sense, e.g. via orthogonal, linear, affine or projective transformations? In other words: so that they are geometrically self-dual?

I would prefer to have a realization of the duality under the weakest possible transformations (i.e. orthogonal), but I wonder whether more general are necessary.


Alathea Jensen defines "self-polar":

Self-polar polytopes are convex polytopes that are equal to an orthogonal transformation of their polar sets.

and writes some interesting things about self-polar polytopes here: https://arxiv.org/abs/1902.00784

Your questions are partially answered there:

  • For all two-dimensional polytopes (a.k.a. polygons) we have (obviously): All self-dual polytopes have self-polar realizations.
  • For all 3-dimensional polytopes the answer is a we have (less obviously, using Koebe-Andreev-Thurston): All self-dual polytopes have self-polar realizations. (This is Theorem 4.6)
  • For arbitrary dimensions this question is still open: "Question 9.2. Does every self-dual polytope have a self-polar realization?"

The paper does not consider affine or projective transformations in general, but does consider the notion "negatively self-polar".

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  • $\begingroup$ Thank you for summarizing the paper! I actually came across this paper quite soon after posting the question. It was interesting to see such a survey of a probably classic topic. To the more general forms of geometric self-duality: I have the feeling, but no concrete arguments, that if a polytope is projectively or affinely self-dual, it is just a matter affine transformation to achieve orthogonal self-duality (self-polarity in the sense of the linked paper). $\endgroup$ – M. Winter Jul 6 '19 at 20:54

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