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We prove that for any dimension d there exists a polynomial time algorithm for counting integral points in polyhedra in the d-dimensional Euclidean space. Previously such algorithms were known for dimensions d = 1, 2, 3, and 4 only.

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A weighted graph is called d-realizable if its vertices can be chosen in d-dimensional Euclidean space so that the Euclidean distance between every pair of adjacent vertices is equal to the prescribed weight. We prove that if a weighted graph with k edges is d-realizable for some d, then it is d-realizable for d = [(8~-k + 1-1)/2] (this bound is sharp in the worst case). We prove that for a graph G with n vertices and k edges and for a dimension d the image of the so-called rigidity map Ea, ~ Rk is a convex set in Ek provided d > [(8v/8-k + 1-1)/2]. These results are obtained as corollaries of a general convexity theorem for quadratic maps which also extends the Toeplitz-Hausdorff theorem. The main ingredients of the proof are the duality for linear programming in the space of quadratic forms and the "corank formula" for the strata of singular quadratic forms. 1. Introduction. Main Results This paper is motivated by the following problem of distance geometry. Suppose that G = (V,E;p) is a graph with the set of vertices V, the set of edges E, and nonnegative weights {Pe: e ~ E} on its edges. We are interested in whether it is possible to place the vertices of G in the Euclidean space R d of a given dimension d so that the Euclidean distance between every pair of adjacent vertices vi, vj would be equal to the prescribed weight Pij. If a map q~: V ~ •d exists such that