L-convex-concave sets in real projective space and L-duality

Abstract

We define a class of L-convex-concave subsets of $\mathbb{R}{P}^n$, where L is a projective subspace of dimension l in $\mathbb{R}{P}^n$. These are sets whose sections by any (l+1)-dimensional space L' containing L are convex and concavely depend on L'. We introduce an L-duality for these sets, and prove that the L-dual to an L-convex-concave set is an $L^{\ast }$-convex-concave subset of $(\mathbb{R}{P}^n{)}^{\ast }$. We discuss a version of Arnold hypothesis for these sets and prove that it is true (or wrong) for an L-convex-concave set and its L-dual simultaneously.

23pp