You are here

Convex pencils of real quadratic forms

TitleConvex pencils of real quadratic forms
Publication TypeJournal Article
Year of Publication2012
AuthorsLerario, A
JournalDiscrete and Computational Geometry, Volume 48, Issue 4, December 2012, Pages 1025-1047

We study the topology of the set X of the solutions of a system of two
quadratic inequalities in the real projective space RP^n (e.g. X is the
intersection of two real quadrics). We give explicit formulae for its Betti
numbers and for those of its double cover in the sphere S^n; we also give
similar formulae 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)\leq 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)\leq 2(k+2).


Sign in