Convex Pencils of Real Quadratic Forms

We study the topology of the set X of the solutions of a system of two quadratic inequalities in the real projective space ?P ⁿ (e.g. X is the intersection of two real quadrics). We give explicit formulas for its Betti numbers and for those of its double cover in the sphere S ⁿ ; we also give similar formulas for level sets of homogeneous quadratic maps to the plane. We discuss some applications of these results, especially in classical convexity theory. We prove the sharp bound b(X)??2n for the total Betti number of X; we show that for odd n this bound is attained only by a singular?X. In the nondegenerate case we also prove the bound on each specific Betti number b _k (X)??2(k+2).