11
$\begingroup$

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.

$\endgroup$
3
$\begingroup$

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".

| cite | improve this answer | |
$\endgroup$
  • $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.