Extremal cross-polytopes and Gaussian vectors

Polytopes for Crystallized Demazure Modules and Extremal Vectors

Demazure’s character formula for arbitrary Kac-Moody Lie algebra was given by S.Kumar and O.Mathieu independently ([6],[8]) by using geometric methods. In 1995, P.Littelmann gave some conjecture (partially solved by himself) about the relation between Demazure’s character formula and crystal bases [7], which was solved affirmatively by M.Kashiwara [3]. Then it gave purely algebraic proof for De...

Extremal Edge Polytopes

The edge polytope of a finite graph G is the convex hull of the columns of its vertex-edge incidence matrix. We study extremal problems for this class of polytopes. For k ∈ {2, 3, 5}, we determine the maximal number of vertices of kneighborly edge polytopes up to a sublinear term. We also construct a family of edge polytopes with exponentially-many facets.

Extremal antipodal polygons and polytopes

5 Let S be a set of 2n points on a circle such that for each point p ∈ S also its antipodal 6 (mirrored with respect to the circle center) point p′ belongs to S. A polygon P of size n is called 7 antipodal if it consists of precisely one point of each antipodal pair (p, p′) of S. 8 We provide a complete characterization of antipodal polygons which maximize (minimize, 9 respectively) the area am...

Autoregressive Gaussian Random Vectors of First Order

Moment Vectors of Polytopes

We give a classification of Borel measurable, SL(d) covariant or contravariant, homogeneous, vector valued valuations on the space of ddimensional convex polytopes containing the origin in their interiors. The only examples are moment vectors of polytopes and moment vectors of polar polytopes. MSC 2000: 52B45, 52A38.