Consider the moment curve in the real Euclidean space R defined parametrically by the map γ : R → R, t → γ(t) = (t, t, . . . , t). The cyclic d-polytope Cd(t1, . . . , tn) is the convex hull of the n, n > d, different points on this curve. The matroidal analogues are the alternating oriented uniform matroids. A polytope [resp. matroid polytope] is called cyclic if its face lattice is isomorphic...

In this note we give the definition of the ”doubling operation” for simple polytopes, find the formula for the h-polynomial of new polytope. As an application of this operation we establish the relationship between moment-angle manifolds and their real analogues and prove the toral rank conjecture for moment-angle manifolds ZP . Let P be a simple n-dimensional polytope with m facets: P = {x ∈ R...

Let O be a symplectic toric orbifold with a fixed T-action and with a toric Kähler metric g. In [10] we explored whether, when O is a manifold, the equivariant spectrum of the Laplace operator ∆g on C∞(O) determines the moment polytope of O, and hence by Delzant’s theorem determines O up to symplectomorphism. In the setting of toric orbifolds we significantly improve upon our previous results a...

We present a general and novel approach for the reconstruction of any convex d-dimensional polytope P , assuming knowledge of finitely many of its integral moments. In particular, we show that the vertices of an N-vertex convex polytope in R can be reconstructed from the knowledge of O(DN) axial moments (w.r.t. to an unknown polynomial measure of degree D), in d + 1 distinct directions in gener...

We show that the multivariate generating function of appropriately normalized moments of a measure with homogeneous polynomial density supported on a compact polytope P ⊂ R is a rational function. Its denominator is the product of linear forms dual to the vertices of P raised to the power equal to the degree of the density function. Using this, we solve the inverse moment problem for the set of...

In this research announcement we associate to each convex polytope, possibly nonrational and nonsimple, a family of compact spaces that are stratified by quasifolds, i.e., the strata are locally modeled by Rk modulo the action of a discrete, possibly infinite, group. Each stratified space is endowed with a symplectic structure and a moment mapping having the property that its image gives the or...

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–...

Dans cette note, on présente l’énoncé et les principales idées de la démonstration d’un théorème de convexité réel pour les applications moment à valeurs dans un groupe de Lie. Ce résultat est un analogue quasi-hamiltonien du théorème de O’Shea et Sjamaar dans le cadre hamiltonien usuel. On démontre que l’image par l’application moment du lieu des points fixes d’une involution renversant la 2-f...

The purpose of this article is to view the Penrose kite from the perspective of symplectic geometry. Mathematics Subject Classification 2000. Primary: 53D20 Secondary: 52C23 Introduction The kite in a Penrose aperiodic tiling by kite and darts [8, 9] is an example of a simple convex polytope. By the Atiyah, Guillemin–Sternberg convexity theorem [1, 6], convex polytopes that are rational can be ...

It is shown that a small cover (resp. real moment-angle manifold) over a simple polytope is an infra-solvmanifold if and only if it is diffeomorphic to a real Bott manifold (resp. flat torus). Moreover, we obtain several equivalent conditions for a small cover being homeomorphic to a real Bott manifold. In addition, we study Riemannian metrics on small covers and real momentangle manifolds with...