Convex Bodies and Algebraic Geometry

Abstract

During the last decade a new area of research has developed relating two subjects which until now had very little in common: convexity and algebraic geometry. An initial success was Stanley's solution of the so-called upper bound conjecture for combinatorial spheres ([16]; for polytopes solved by McMullen, published in [14]): Among all combinatorial spheres with v vertices (faces are convex polytopes), the convex hulls of v points on a moment curve {(t, t2, ..., t")lt E R} (called cyclic polytopes) possess maximal numbers of faces in all dimensions. This result has been pursued further by Kind and Kleinschmidt [lo]. Another result is the solution of McMullen's conjecture about characterizing those vectors f (P) = ( fo(P), . . . , fd-l(P)) for which f'P) is the number ofj-faces of a polytope P. The necessity of McMullen's condition has been shown by Stanley [17] and the sufficiency by Billera and Lee [2]. A foundation for Stanley's, Billera's, and Lee's work was laid by Hochster [S].This paper by Hochster was also one of the starting points for a development which is to be outlined in what follows. We emphasize two main achievements: First, a characterization of Milnor's number of critical points of complex algebraic functions by numbers assigned to convex polytopes (Kouchnirenko [ll]); second, characterizations of mixed volumes of convex bodies as the intersection index of certain varieties, and an alternative proof of the Alexandrov-Fenchel inequality (Bernstein [l], Burago and Zalgaller [4], Teissier [19]). In order to make the exposition comprehensible to nonspecialists, we restrict the discussion to elementary explanations.