Mathematics

In this paper we discuss three distance functions on the set of convex bodies. In particular we study the convergence of Delzant polytopes, which are fundamental objects in symplectic toric geometry. By using these observations, we derive some convergence theorems for symplectic toric manifolds with respect to the Gromov-Hausdorff distance.

Let S be a set of points in R 2 contained in a circle and P an unrestricted point set in R 2 . We prove the number of distinct distances between points in S and points in P is at least min(|S||P | 1/4−ε ,|S | 2/3 |P | 2/3 ,|S | 2 ,|P | 2 ) . This builds on work of Pach and De Zeeuw, Bruner and Sharir, McLaughlin and Omar and Mathialagan on distances between pairs of sets.

We compute the distortion coefficients of the α -Grushin plane. They are expressed in terms of generalised trigonometric functions. Estimates for the distortion coefficients are then obtained and a conjecture of a curvature-dimension condition for the generalised Grushin planes is suggested.

We prove that in arbitrary Carnot groups G of step 2, with a splitting G=W⋅L with L one-dimensional, the graph of a continuous function φ:U⊆W→L is C 1 H -regular precisely when φ satisfies, in the distributional sense, a Burgers' type system D φ φ=ω , with a continuous ω . We stress that this equivalence does not hold already in the easiest step-3 Carnot group, namely the Engel group. As a tool for the proof we show that a continuous distributional solution φ to a Burgers' type system D φ φ=ω , with ω continuous, is actually a broad solution to D φ φ=ω . As a by-product of independent interest we obtain that all the continuous distributional solutions to D φ φ=ω , with ω continuous, enjoy 1/2 -little Hölder regularity along vertical directions.

We introduce a hierachy of equivalence relations on the set of separated nets of a given Euclidean space, indexed by concave increasing functions ?:(0,????0,?? . Two separated nets are called ? -displacement equivalent if, roughly speaking, there is a bijection between them which, for large radii R , displaces points of norm at most R by something of order at most ?(R) . We show that the spectrum of ? -displacement equivalence spans from the established notion of bounded displacement equivalence, which corresponds to bounded ? , to the indiscrete equivalence relation, coresponding to ?(R)?��?R) , in which all separated nets are equivalent. In between the two ends of this spectrum, the notions of ? -displacement equivalence are shown to be pairwise distinct with respect to the asymptotic classes of ?(R) for R?��? . We further undertake a comparison of our notion of ? -displacement equivalence with previously studied relations on separated nets. Particular attention is given to the interaction of the notions of ? -displacement equivalence with that of bilipschitz equivalence.

For an r -tuple ( γ 1 ,…, γ r ) of special orthogonal d×d matrices, we say the Euclidean (d−1) -dimensional sphere S d−1 is ( γ 1 ,…, γ r ) -divisible if there is a subset A⊆ S d−1 such that its translations by the rotations γ 1 ,…, γ r partition the sphere. Motivated by some old open questions of Mycielski and Wagon, we investigate the version of this notion where the set A has to be measurable with respect to the spherical measure. Our main result shows that measurable divisibility is impossible for a "generic" (in various meanings) r -tuple of rotations. This is in stark contrast to the recent result of Conley, Marks and Unger which implies that, for every "generic" r -tuple, divisibility is possible with parts that have the property of Baire.

3D-facets of the Delone cells representing the deep and shallow holes of the root lattice D6 which tile the six-dimensional Euclidean space in an alternating order are projected into three-dimensional space. They are classified into six Mosseri-Sadoc tetrahedral tiles of edge lengths 1 and golden ratio (tau) with faces normal to the 5-fold and 3-fold axes. The icosahedron, dodecahedron and icosidodecahedron whose vertices are obtained from the fundamental weights of the icosahedral group are dissected in terms of six tetrahedra. A set of four tiles are composed out of six fundamental tiles, faces of which, are normal to the 5-fold axes of the icosahedral group. It is shown that the 3D-Euclidean space can be tiled face-to-face with maximal face coverage by the composite tiles with an inflation factor tau generated by an inflation matrix. We note that dodecahedra with edge lengths of 1 and tau naturally occur already in the second and third order of the inflations. The 3D patches displaying 5-fold, 3-fold and 2-fold symmetries are obtained in the inflated dodecahedral structures with edge lengths tau to the power n with n equals 3 or greater than 3. The planar tiling of the faces of the composite tiles follow the edge-to-edge matching of the Robinson triangles.

Let K be a compact convex body in R n . For any affine line L, denote ? ? K (L)= ??L ? K (x)dl(x), where dl is the arc length measure, the X -ray transform of the characteristic function ? K , i.e., the length of the chord K?�L. We prove that if K is bounded by a C ??real algebraic hypersurface ?�K and the X -ray transform ? ? K (L) behaves, under small parallel translations of the line L to the distance t, as the m -th root of a polynomial of t , for some fixed m?�N, then ?�K is an ellipsoid.

A closed PL-curve is called integral if it is comprised of unit intervals. Kenyon's problem asks whether for every integral curve γ in R 3 , there is a dome over γ , i.e. whether γ is a boundary of a polyhedral surface whose faces are equilateral triangles with unit edge lengths. First, we give an algebraic necessary condition when γ is a quadrilateral, thus giving a negative solution to Kenyon's problem in full generality. We then prove that domes exist over a dense set of integral curves. Finally, we give an explicit construction of domes over all regular n -gons.

We generalize the recent results of Björn-Björn-Shanmugalingam \cite{BBS20} concerning how measures transform under the uniformization procedure of Bonk-Heinonen-Koskela for Gromov hyperbolic spaces \cite{BHK} by showing that these results also hold in the setting of uniformizing Gromov hyperbolic spaces by Busemann functions that we introduced in \cite{Bu20}. In particular uniformly local doubling and uniformly local Poincaré inequalities for the starting measure transform into global doubling and global Poincaré inequalities for the uniformized measure. We then show in the setting of uniformizations of universal covers of closed negatively curved Riemannian manifolds equipped with the Riemannian measure that one can obtain sharp ranges of exponents for the uniformized measure to be doubling and satisfy a 1 -Poincaré inequality. Lastly we introduce the procedure of uniform inversion for uniform metric spaces, and show that both the doubling property and the p -Poincaré inequality are preserved by uniform inversion for any p?? .