Note on the Complexity of the Mixed-Integer Hull of a Polyhedron

We study the complexity of computing the mixed-integer hull $\operatorname{conv}(P\cap\mathbb{Z}^n\times\mathbb{R}^d)$ of a polyhedron $P$. Given an inequality description, with one integer variable, the mixed-integer hull can have exponentially many vertices and facets in $d$. For $n,d$ fixed, we give an algorithm to find the mixed integer hull in polynomial time. Given $P=\operatorname{conv}(V)$ and $n$ fixed, we compute a vertex description of the mixed-integer hull in polynomial time and give bounds on the number of vertices of the mixed integer hull

Embedding SnS^n into Rn+1R^{n+1} with given integral Gauss curvature and optimal mass transport on SnS^n

In his book on Convex Polyhedra (section 7.2), A.D. Aleksandrov raised a general question of finding variational statements and proofs of existence of polytopes with given geometric data. The first goal of this paper is to give a variational solution to the problem of existence and uniqueness of a closed convex hypersurface in Euclidean space with prescribed integral Gauss curvature. Our solution includes the case of a convex polytope. This problem was also first considered by Aleksandrov and below it is referred to as Aleksandrov's problem. The second goal of this paper is to show that in variational form the Aleksandrov problem is closely connected with the theory of optimal mass transport on a sphere with cost function and constraints arising naturally from geometric considerations

Global residues for sparse polynomial systems

We consider families of sparse Laurent polynomials f_1,...,f_n with a finite set of common zeroes Z_f in the complex algebraic n-torus. The global residue assigns to every Laurent polynomial g the sum of its Grothendieck residues over the set Z_f. We present a new symbolic algorithm for computing the global residue as a rational function of the coefficients of the f_i when the Newton polytopes of the f_i are full-dimensional. Our results have consequences in sparse polynomial interpolation and lattice point enumeration in Minkowski sums of polytopes.Comment: Typos corrected, reference added, 13 pages, 5 figures. To appear in JPA