# Geometric realization of combinatorial self-duality in polytopes

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