  We define a class of L-convex-concave subsets of $\Bbb{R}P^n$, where L is a
projective subspace of dimension l in $\Bbb{R}P^n$. 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 $L^*$-convex-concave subset of
$(\Bbb RP^n)^*$. 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.
