THE HYPERMETRIC CONE IS POLYHEDRAL

The hypermetric cone H(n) is the cone in the space R(n(n-1)/2) of all vectors d=(d(ij))1 less-than-or-equal-to i < j less-than-or-equal-to n satisfying the hypermetric inequalities: SIGMA1 less-than-or-equal-to i < j less-than-or-equal-to n z(j)z(j)d(ij) less-than-or-equal-to 0 for all integer vectors z in Z(n) with SIGMA1 less-than-or-equal-to i less-than-or-equal-to n z(i) = 1. We explore connections of the hypermetric cone with quadratic forms and the geometry of numbers (empty spheres and L-polytopes in lattices). As an application, we show that the hypermetric cone H(n) is polyhedral.