Abstract: In this paper, we introduce the notion of augmentation for polytopes and use it to show the error in two presumptions that have been key in arriving at over-reaching/over-scoped claims of “impossibility” in recent extended formulations (EF) developments. One of these presumption is that: “If Polytopes P and Qare described in the spaces of variables xand yrespectively, and there exists...

Polytope approach of genetic immunization is a promising strategy for the prevention of infectious disease as it is capable of generating effective cell mediated immunity by delivering the T cell epitopes assembled in series. Leishmaniasis is a significant world wide health problem for which no vaccine exists. In this study we have compared immunogenicity and efficacy of three types of DNA vacc...

The max-cut and stable set problems are two fundamental NP -hard problems in combinatorial optimization. It has been known for a long time that any instance of the stable set problem can be easily transformed into a max-cut instance. Moreover, Laurent, Poljak, Rendl and others have shown that any convex set containing the cut polytope yields, via a suitable projection, a convex set containing t...

A code polytope is defined to be the convex hull in R n of the points in {0, 1}n corresponding to the codewords of a binary linear code. This paper contains a collection of results concerning the structure of such code polytopes. A survey of known results on the dimension and the minimal polyhedral representation of a code polytope is first presented. We show how these results can be extended t...

An action of a torus T on a compact connected symplectic manifold M is Hamiltonian if it admits a momentum map, which is a map on M taking values in the dual of the Lie algebra of T. In 1982, Atiyah and GuilleminSternberg proved that the image of the momentum map is a convex polytope, now called the momentum polytope. In 1988, Delzant showed that if the dimension of T is half of the dimension o...

This note studies the geometric structure of monotone moment polytopes (the duals of smooth Fano polytopes) using probes. The latter are line segments that enter the polytope at an interior point of a facet and whose direction is integrally transverse to this facet. A point inside the polytope is displaceable by a probe if it lies less than half way along it. Using a construction due to Fukaya–...

This note summarizes recent results from computational geometry which determine complexity of computing Minkowski sum of k convex polytopes in R, which are represented either in terms of facets or in terms of vertices. In particular, it is pointed out for which cases there exists an algorithm which runs in polynomial time. The note is based on papers of Gritzmann and Sturmfels [6] and Komei Fuk...