### A Note on Geometry of Polyhedron of Three-index Transportation Problem

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Given a planar convex body B centered at the origin, we denote by ℳ ²(B) the Minkowski plane (i.e., two-dimensional linear normed space) with the unit ball B. For a triangle T in ℳ ²(B) we denote by ${R}_{B}\left(T\right)$ the least possible radius of a Minkowskian ball enclosing T. We remark that in the terminology of location science ${R}_{B}\left(T\right)$ is the optimum of the minimax location problem with distance induced by B and vertices of T as existing facilities (see, for instance, [HM03] and the references therein). Using methods...

En dos artículos, publicados en 1989, Balas y Ng dan una metodología para construir facetas del politopo de recubrimiento con coeficientes en {0, 1, 2}. Siguiendo esta metodología, en el presente artículo decimos cómo se contruyen facetas de dicho politopo con coeficientes en {0, 1, 2, 3}.

The Löwner-John ellipse of a full-dimensional bounded convex set is a circumscribed ellipse with the property that if we shrink it by the factor $n$ (where $n$ is dimension), we obtain an inscribed ellipse. Goffin’s algorithm constructs, in polynomial time, a tight approximation of the Löwner-John ellipse of a polyhedron given by facet description. In this text we adapt the algorithm for zonotopes given by generator descriptions. We show that the adapted version works in time polynomial in the size...

Through tropical normal idempotent matrices, we introduce isocanted alcoved polytopes, computing their $f$-vectors and checking the validity of the following five conjectures: Bárány, unimodality, ${3}^{d}$, flag and cubical lower bound (CLBC). Isocanted alcoved polytopes are centrally symmetric, almost simple cubical polytopes. They are zonotopes. We show that, for each dimension, there is a unique combinatorial type. In dimension $d$, an isocanted alcoved polytope has ${2}^{d+1}-2$ vertices, its face lattice is the lattice...

We study the problem of finding the smallest $m$ such that every element of an exponential family can be written as a mixture of $m$ elements of another exponential family. We propose an approach based on coverings and packings of the face lattice of the corresponding convex support polytopes and results from coding theory. We show that $m={q}^{N-1}$ is the smallest number for which any distribution of $N$$q...$