We consider the semigroup $S$ of highest weights appearing in tensor powers $V^{otimes k}$ of a finite dimensional representation $V$ of a connected reductive group. We describe the cone generated by $S$ as the cone over the weight polytope of $V$ intersected with the positive Weyl chamber. From this we get a description for the asymptotic of the number of highest weights appearing in $V^{otime...

Since at least half of the d edges incident to a vertex u of a simple d-polytope P either all point “up” or all point “down,” v must be the unique “bottom” or “top” vertex of a face of P of dimension at least d/2. Thus the number of P’s vertices is at most twice the number of such high-dimensional faces, which is at most Ed,2 $ iG &Ynu) = O(nld/‘l), if P has n facets. This, in a nutshell, provi...

where the indexing set Fatg,n is the space of labeled fatgraphs of genus g and n boundary components. See Section 2 for definitions of a fatgraph Γ, its automorphism group AutΓ and the cell decomposition (1) realised as the space of labeled fatgraphs with metrics. Restricting this homeomorphism to a fixed n-tuple of positive numbers (b1, ..., bn) yields a space homeomorphic to Mg,n decomposed i...

We show how to realize a stacked 3D polytope (formed by repeatedly stacking a tetrahedron onto a triangular face) by a strictly convex embedding with its n vertices on an integer grid of size O(n)× O(n)× O(n). We use a perturbation technique to construct an integral 2D embedding that lifts to a small 3D polytope, all in linear time. This result solves a question posed by Günter M. Ziegler, and ...

The aim of this paper is the determination of the largest n-dimensional polytope with n+3 vertices of unit diameter. This is a special case of a more general problem Graham proposes in [2].

This paper gives an algorithm for polytope covering : let L and U be sets of points in R, comprising n points altogether. A cover for L from U is a set C ⊂ U with L a subset of the convex hull of C. Suppose c is the size of a smallest such cover, if it exists. The randomized algorithm given here finds a cover of size no more than c(5d ln c), for c large enough. The algorithm requires O(cn) expe...

Let u and v be permutations on n letters, with u ≤ v in Bruhat order. A Bruhat interval polytope Qu,v is the convex hull of all permutation vectors z = (z(1), z(2), . . . , z(n)) with u ≤ z ≤ v. Note that when u = e and v = w0 are the shortest and longest elements of the symmetric group, Qe,w0 is the classical permutohedron. Bruhat interval polytopes were studied recently in the 2013 paper “The...

The volume of the n-dimensional polytope n(x) := fy 2 R n : yi 0 and y1 + + yi x1 + + xi for all 1 i ng for arbitrary x := (x1; : : : ; xn) with xi > 0 for all i de nes a polynomial in variables xi which admits a number of interpretations, in terms of empirical distributions, plane partitions, and parking functions. We interpret the terms of this polynomial as the volumes of chambers in two di ...

We prove that any extended formulation that approximates the matching polytope on nvertex graphs up to a factor of (1 + ε) for any 2 n ≤ ε ≤ 1 must have at least ( n α/ε ) defining inequalities where 0 < α < 1 is an absolute constant. This is tight as exhibited by the (1 + ε) approximating linear program obtained by dropping the odd set constraints of size larger than (1 + ε)/ε from the descrip...