The cross covariogram of a pair of polygons determines both polygons, with a few exceptions

The cross covariogram g_{K,L} of two convex sets K and L in R^n is the function which associates to each x in R^n the volume of the intersection of K and L+x. Very recently Averkov and Bianchi [AB] have confirmed Matheron's conjecture on the covariogram problem, that asserts that any planar convex body K is determined by the knowledge of g_{K,K}. The problem of determining the sets from their covariogram is relevant in probability, in statistical shape recognition and in the determination of the atomic structure of a quasicrystal from X-ray diffraction images. We prove that when K and L are convex polygons (and also when K and L are planar convex cones) g_{K,L} determines both K and L, up to a described family of exceptions. These results imply that, when K and L are in these classes, the information provided by the cross covariogram is so rich as to determine not only one unknown body, as required by Matheron's conjecture, but two bodies, with a few classified exceptions. These results are also used by Bianchi [Bia] to prove that any convex polytope P in R^3 is determined by g_{P,P}.Comment: 26 pages, 9 figure

Tropical Severi Varieties

We study the tropicalizations of Severi varieties, which we call tropical Severi varieties. In this paper, we give a partial answer to the following question, ``describe the tropical Severi varieties explicitly.'' We obtain a description of tropical Severi varieties in terms of regular subdivisions of polygons. As an intermediate step, we construct explicit parameter spaces of curves. These parameter spaces are much simpler objects than the corresponding Severi variety and they are closely related to flat degenerations of the Severi variety, which in turn describes the tropical Severi variety. As an application, we understand G.Mikhalkin's correspondence theorem for the degrees of Severi varieties in terms of tropical intersection theory. In particular, this provides a proof of the independence of point-configurations in the enumeration of tropical nodal curves.Comment: 25 pages, Final version accepted to Portugal. Mat

Symmetries of Monocoronal Tilings

The vertex corona of a vertex of some tiling is the vertex together with the adjacent tiles. A tiling where all vertex coronae are congruent is called monocoronal. We provide a classification of monocoronal tilings in the Euclidean plane and derive a list of all possible symmetry groups of monocoronal tilings. In particular, any monocoronal tiling with respect to direct congruence is crystallographic, whereas any monocoronal tiling with respect to congruence (reflections allowed) is either crystallographic or it has a one-dimensional translation group. Furthermore, bounds on the number of the dimensions of the translation group of monocoronal tilings in higher dimensional Euclidean space are obtained.Comment: 26 pages, 66 figure

Veech surfaces and simple closed curves

We study the SL(2,R)-infimal lengths of simple closed curves on half-translation surfaces. Our main result is a characterization of Veech surfaces in terms of these lengths. We also revisit the "no small virtual triangles" theorem of Smillie and Weiss and establish the following dichotomy: the virtual triangle area spectrum of a half-translation surface either has a gap above zero or is dense in a neighborhood of zero. These results make use of the auxiliary polygon associated to a curve on a half-translation surface, as introduced by Tang and Webb.Comment: 12 pages. v2: added proof of continuity of infimal length functions on quadratic differential space; 16 pages, one figure; to appear in Israel J. Mat