Mathematics

In this note, we present a unified approach to the problem of existence of a potential for the optimal transport problem with respect to non-traditional cost functions, that is costs that assume infinite values. We establish a new method which relies on proving solvability of a special (possibly infinite) family of linear inequalities. We give a necessary and sufficient condition on the coefficients that assure the existence of a solution, and which in the setting of transport theory we call c -path-boundedness. This condition fully characterizes sets that admit a potential and replaces c -cyclic monotonicity from the classical theory, i.e. when the cost is real-valued. Our method also gives a new and elementary proof for the classical results of Rockafellar, Rochet and Rüschendorf.

Start with Gott (2019)'s envelope polyhedron (Squares-4 around a point): a unit cube missing its top and bottom faces. Stretch by a factor of 2 in the vertical direction so its sides become (2x1 unit) rectangles. This has 8 faces (4 exterior, 4 interior), 8 vertices, and 16 edges. F-E+V = 0, implying a (toroidal) genus = 1. It is isometric to a flat square torus. Like any polyhedron it has zero intrinsic Gaussian curvature on its faces and edges. Since 4 right angled rectangles meet at each vertex, there is no angle deficit and zero Gaussian curvature there as well. All meridian and latitudinal circumferences are equal (4 units long).

We introduce new definitions of sectional, Ricci and scalar curvature for networks and their higher dimensional counterparts, derived from two classical notions of curvature for curves in general metric spaces, namely, the Menger curvature and the Haantjes curvature. These curvatures are applicable to unweighted or weighted and undirected or directed networks, and are more intuitive and easier to compute than other network curvatures. In particular, the proposed curvatures based on the interpretation of Haantjes definition as geodesic curvature allow us to give a network analogue of the classical local Gauss-Bonnet theorem. Furthermore, we propose even simpler and more intuitive proxies for the Haantjes curvature that allow for even faster and easier computations in large-scale networks. In addition, we also investigate the embedding properties of the proposed Ricci curvatures. Lastly, we also investigate the behaviour, both on model and real-world networks, of the curvatures introduced herein with more established notions of Ricci curvature and other widely-used network measures.

We consider three classes of geodesic embeddings of graphs on Euclidean flat tori: (1) A torus graph G is equilibrium if it is possible to place positive weights on the edges, such that the weighted edge vectors incident to each vertex of G sum to zero. (2) A torus graph G is reciprocal if there is a geodesic embedding of the dual graph G ∗ on the same flat torus, where each edge of G is orthogonal to the corresponding dual edge in G ∗ . (3) A torus graph G is coherent if it is possible to assign weights to the vertices, so that G is the (intrinsic) weighted Delaunay graph of its vertices. The classical Maxwell-Cremona correspondence and the well-known correspondence between convex hulls and weighted Delaunay triangulations imply that the analogous concepts for plane graphs (with convex outer faces) are equivalent. Indeed, all three conditions are equivalent to G being the projection of the 1-skeleton of the lower convex hull of points in R 3 . However, this three-way equivalence does not extend directly to geodesic graphs on flat tori. On any flat torus, reciprocal and coherent graphs are equivalent, and every reciprocal graph is equilibrium, but not every equilibrium graph is reciprocal. We establish a weaker correspondence: Every equilibrium graph on any flat torus is affinely equivalent to a reciprocal/coherent graph on some flat torus.

Let O be a closed n -dimensional arithmetic (real or complex) hyperbolic orbifold. We show that the diameter of O is bounded above by c 1 logvol(O)+ c 2 h(O) , where h(O) is the Cheeger constant of O , vol(O) is its volume, and constants c 1 , c 2 depend only on n .

A cap of spherical radius α on a unit d -sphere S is the set of points within spherical distance α from a given point on the sphere. Let F be a finite set of caps lying on S . We prove that if no hyperplane through the center of S divides F into two non-empty subsets without intersecting any cap in F , then there is a cap of radius equal to the sum of radii of all caps in F covering all caps of F provided that the sum of radii is less π/2 . This is the spherical analog of the so-called Circle Covering Theorem by Goodman and Goodman and the strengthening of Fejes Tóth's zone conjecture proved by Jiang and the author arXiv:1703.10550.

Let K be a unit ball of some norm in R n . For an arbitrary direction u∈ R n , there is associated a unit-ball K u , which is rotationally invariant with respect to rotations keeping u fixed, called the u -spin of K u . It is proved that K is a zonoid if and only if all of its spins are zonoids.

We improve a lower bound for the smallest area of convex covers for closed unit curves from 0.0975 to 0.1, which makes it substantially closer to the current best upper bound 0.11023. We did this by considering the minimal area of convex hull of circle, line of length 1/2, and rectangle with side 0.1727 x 0.3273. By using geometric methods and the Box search algorithm, we proved that this area is at least 0.1. We give informal numerical evidence that the obtained lower bound is close to the limit of current techniques, and substantially new idea is required to go significantly beyond 0.10044.

We prove that in every compact space of Delone sets in R d which is minimal with respect to the action by translations, either all Delone sets are uniformly spread, or continuously many distinct bounded displacement equivalence classes are represented, none of which contains a lattice. The implied limits are taken with respect to the Chabauty--Fell topology, which is the natural topology on the space of closed subsets of R d . This topology coincides with the standard local topology in the finite local complexity setting, and it follows that the dichotomy holds for all minimal spaces of Delone sets associated with well-studied constructions such as cut-and-project sets and substitution tilings, whether or not finite local complexity is assumed.

We review the theory of Gradient Flows in the framework of convex and lower semicontinuous functionals on CAT(κ) -spaces and prove that they can be characterized by the same differential inclusion y ??t ?��? ????E( y t ) one uses in the smooth setting and more precisely that y ??t selects the element of minimal norm in ??????E( y t ) . This generalizes previous results in this direction where the energy was also assumed to be Lipschitz. We then apply such result to the Korevaar-Schoen energy functional on the space of L 2 and CAT(0) valued maps: we define the Laplacian of such L 2 map as the element of minimal norm in ??????E(u) , provided it is not empty. The theory of gradient flows ensures that the set of maps admitting a Laplacian is L 2 -dense. Basic properties of this Laplacian are then studied.