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

الأصل · EN

We define a class of L-convex-concave subsets of RPⁿ, where L is a projective subspace of dimension l in RPⁿ. 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*-convex-concave subset of (RPⁿ)*. 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.