L-convex-concave Sets in Real Projective Space and L-duality*

Convex-concave sets and Arnold hypothesis. The notion of convexity is usually defined for subsets of affine spaces, but it can be generalized for subsets of projective spaces. Namely, a subset of a projective space RP is called convex if it doesn’t intersect some hyperplane L ⊂ RP and is convex in the affine space RP \L. In the very definition of the convex subset of a projective space appears a hyperplane L. In projective space there are subspaces L of different dimensions, not only hyperplanes. For any subspace L one can define a class of L-convex-concave sets. These sets are the main object of investigation in this paper. If L is a hyperplane then this class coincides with the class of closed convex sets lying in the affine chart RP \ L. Here is the definition of L-convex-concave sets. A closed set A ⊂ RP is L-convex-concave if: 1) the set A doesn’t intersect the projective subspace L, 2) for any (dim L + 1)-dimensional subspace N ⊂ RP containing L the section A∩N of the set A by N is convex, 3) for any (dim L− 1)dimensional subspace T ⊂ L the complement to the projection of the set A from the center T on the factor-space RP/T is an open convex set. Example. In a projective space RP with homogeneous coordinates x0 : · · · : xn one can consider a set A ⊂ RP defined by the inequality {K(x) ≤ 0}, where K is a non-degenerate quadratic form on R. Suppose that K is positively defined on some (k + 1)-dimensional subspace, and is negatively defined on some (n − k)dimensional subspace. In other words, suppose that (up to a linear change of coordinates) the form K is of the form K(x) = x0 + · · ·+ xk − xk+1 − · · · − xn. In this case the set A is L-convex-concave with respect to projectivization L of any (k + 1)-dimensional subspace of R on which K is positively defined.