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We define the alternating sign matrix polytope as the convex hull of $n\times n$ alternating sign matrices and prove its equivalent description in terms of inequalities. This is analogous to the well known result of Birkhoff and von Neumann that the convex hull of the permutation matrices equals the set of all nonnegative doubly stochastic matrices. We count the facets and vertices of the alternating sign matrix polytope and describe its projection to the permutohedron as well as give adoi:10.37236/130 fatcat:fy27abyz7bae7hkmh3rbvso4ea