Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0805.1805

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

Gabriele Bianchi
View original source ↗Read source document ↗Export metadata ↓

Abstract

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}.

Explore related subjects

math.MGmath.STstat.TH

Keep this discovery

BibTeXRIS

Gabriele Bianchi. 2008-05-13. The cross covariogram of a pair of polygons determines both polygons, with a few exceptions. https://doi.org/10.1016/j.aam.2008.10.002

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Tile sets consisting of two types of concave polygons derived from periodic tilings corresponding to non-periodic tilings with hat and turtle tiles

Using a convex pentagonal monotile belonging to the Type 5 family, we investigate the relationships among the hat tile, turtle tile, and Tile$(1, 1)$. By applying Sugimoto's Perspective and Amfirifma's Perspective, we obtain four types of concave polygons, AH-tile, BH-tile, AT-tile, and BT-tile, each having Heesch number 1 under the conditions considered. We show that these polygons correspond to clusters used to generate the non-periodic tilings $\mathscr{T}_h$ and $\mathscr{T}_s$. We further discuss the possibility that tile sets consisting of pairs selected from these polygons may correspond to $\textit{ASPmr}\{\text{A-tile}, \text{B-tile}\}$.

math.MG↗
arXiv2026

The mean distance to a simple closed curve on the sphere

Kimberling's Problem 10 asks for a simple closed curve of prescribed length $L$ (in particular, $L=4\pi$) on the unit sphere minimizing the mean geodesic distance $\mathcal{J}$ from a point of the sphere to the curve. For a positive integer $n$, put $\vartheta_{n}=\pi/(2n)$ and $L_{n}=2\pi/\sin\vartheta_{n}$. We show that the minimum of $\mathcal{J}$ over rectifiable simple closed curves of length at most $L_{n}$ equals $\vartheta_{n}-\tan(\vartheta_{n}/2)$, that it is attained only by curves of length exactly $L_{n}$, and that the sphere-filling ropes $\beta^{n,k}$ of Gerlach and von der Mosel attain it. Kimberling's case is $n=3$: at $L=4\pi$ the minimum is $\pi/6+\sqrt{3}-2=0.255649\ldots$, attained by an explicit six-arc curve and by its mirror image. For $L\le2\pi$ we determine $J(L)$, the infimum of $\mathcal{J}$ over curves of length $L$, exactly: it equals $\pi/2-L/(2\pi)$, attained precisely by the circles of length $L$. At the lengths $L_{n}$ we do not classify all minimizers, but show that every one of them bisects the sphere into two disks of area $2\pi$ and inradius $\vartheta_{n}$ whose inward collars have the largest possible area at every depth. The great circle is the only minimizer for $n=1$, and the $\beta^{n,k}$ are, up to congruence, the only ones of thickness at least $\sin\vartheta_{n}$. For arbitrary $L$ the function $J$ is nonincreasing, and together with the above this brackets it between two explicit values.

math.MG↗
arXiv2026

The topology of Gromov--Hausdorff space

We prove that the Gromov--Hausdorff space is homeomorphic to the Hilbert space. This paper is divided into four parts. In Part I, we construct an assignment of a full-support probability measure to every nonempty compact metric space that respects isometries and is continuous for simultaneous Hausdorff convergence of the spaces and weak convergence of the measures. In Part II, we use these measures to construct finite-dimensional local models whose induced pseudometrics approximate the original distances uniformly and whose norms and point maps vary continuously up to orthogonal changes of coordinates. In Part III, we use the local models to prove that the Gromov--Hausdorff space is an absolute retract for all metrizable spaces. In Part IV, we establish a discrete approximation property and conclude that the space of isometry classes of nonempty compact metric spaces is homeomorphic to the real separable infinite-dimensional Hilbert space.

math.MG↗