Self-polar polytopes

Self-dual polyhedral cones and their slack matrices

“…Their study goes back to seminal works of Lovász [Lov83] and Grünbaum [GS88] and are an integral part of a more general body of work on self-duality of polytopes. In [Jen21] one can find a modern survey on the area from this perspective, that revisits many of the results from [BF76]. In order to translate our previous results, one needs to adapt some of the definitions to this context.…”

“…which is self-dual ( [TE14,Jen21]). This is obtained by applying the second construction described in the proof of [Jen21,Theorem 4.7] to the case k = 1 to generate a negatively self-polar polytope P ⊆ R 3 .…”

“…which is self-dual ( [TE14,Jen21]). This is obtained by applying the second construction described in the proof of [Jen21,Theorem 4.7] to the case k = 1 to generate a negatively self-polar polytope P ⊆ R 3 . Then, proceeding as in Section 3.3, the corresponding cone in R 4 generated by 1 × P is self-dual and has the extreme rays as in (4.3).…”

“…The proof of Proposition 4.13 in the paper [Jen21] is completely constructive, so we can actually construct these sandwiched cones explicitly, and therefore automatically construct the matrices M and B. In practice, however, there seem to be simpler ways of finding ad hoc contructions for simple examples of extreme rays of DNN n that are not slacks.…”

“…These correspond to polyhedral cones of dimensions four and five respectively. Note that the case for three dimensional polyhedral cones was settled by [BF76]: they have self-dual realizations if and only if they have an odd number of rays, which happens if and only if they are strongly involutive self-dual (see also [Jen21,Corollary 4.2]). Here, by a self-dual realization of a cone K we mean a self-dual cone K ′ that is combinatorially equivalent to K.…”