In the last decade, tropical geometry has been attracted a lot of attention in various fields such as the algebraic geometry, computational biology, and physics. The tropical polytope in the tropical geometry was introduced by Develin and Sturmfels as a counterpart of the polytope in the (ordinal) geometry. Recently, in the theory of directed multiflows, it has been shown by Hirai and the autho...

The contact polytope of a lattice is the convex hull of its shortest vectors. In this paper we classify the facets of the contact polytope of the Leech lattice up to symmetry. There are 1,197,362,269,604,214,277,200 many facets in 232 orbits.

The hyperplane separation bound is a lower on the extension complexity of polytope. It main tool in Rothvoß's proof an exponential for matching polytope (Rothvoß, 2017). We show that technique sensitive to choice slack matrix and does not improve upon best known bounds spanning tree completion time polytopes when applied their canonical matrices. Stronger may be obtained by appropriate rescalin...

We begin with a lattice N isomorphic to Z. The dual lattice M of N is given by Hom(N,Z); it is also isomorphic to Z. (The alphabet may appear to be going backwards; but this notation is standard in the literature.) We write the pairing of v ∈ N and w ∈M as 〈v, w〉. A cone in N is a subset of the real vector space NR = N ⊗R generated by nonnegative R-linear combinations of a set of vectors {v1, ....

In [1], the author generalized Ehrhart’s idea ([2]) of counting lattice points in dilated rational polytopes: Given a rational polytope, that is, a polytope with rational vertices, we use its description as the intersection of halfspaces, which determine the facets of the polytope. Instead of just a single dilation factor, we allow different dilation factors for each of these facets. We proved ...