Mathematics

We prove that the volume of central hyperplane sections of a unit cube in R n orthogonal to a diameter of the cube is a strictly monotonically increasing function of the dimension for n≥3 . Our argument uses an integral formula that goes back to Pólya \cite{P} (see also \cite{H} and \cite{B86}) for the volume of central sections of the cube, and Laplace's method to estimate the asymptotic behaviour of the integral. First we show that monotonicity holds starting from some specific n 0 . Then, using interval arithmetic (IA) and automatic differentiation (AD), we compute an explicit bound for n 0 , and check the remaining cases between 3 and n 0 by direct computation.

In this work we prove that either a sequence of axis of symmetry or a sequence of hyperplanes of symmetry of a convex body K in the Euclidian space E d ,n≥3 , are enough to guarantee that K is either a generalized body of revolution or a sphere.

In this work we prove that if there exists a smooth convex body M in the Euclidean space R n , n≥3 , contained in the interior of the unit ball S n−1 of R n , and point p∈ R n such that, for each point of S n−1 , M looks centrally symmetric and p appears as the centre, then M is an sphere.

In arbitrary Carnot groups we study intrinsic graphs of maps with horizontal target. These graphs are C 1 H regular exactly when the map is uniformly intrinsically differentiable. Our first main result characterizes the uniformly intrinsic differentiability by means of Hölder properties along the projections of left-invariant vector fields on the graph. We strengthen the result in step-2 Carnot groups for intrinsic real-valued maps by only requiring horizontal regularity. We remark that such a refinement is not possible already in the easiest step-3 group. As a by-product of independent interest, in every Carnot group we prove an area-formula for uniformly intrinsically differentiable real-valued maps. We also explicitly write the area element in terms of the intrinsic derivatives of the map.

Koskela and Zhou have proven that, on the harmonic Sierpinski gasket with Kusuoka's measure, the "natural" Dirichlet form coincides with Cheeger's energy. We give a different proof of this result, which uses the properties of the Lyapounov exponent of the gasket.

We prove that Cheng maximal diameter theorem for hypergraphs with positive coarse Ricci curvature.

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a continuous map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve the quasi-isometry constants provided by Biswas in the case of CAT(-1)} spaces. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces.

We give characterizations for the parabolicity of regular trees.

We investigate the behavior of a complete flat metric on a surface near a puncture. We call a puncture on a flat surface regular if it has a neighborhood which is isometric to that of a point at infinity of a cone. We prove that there are punctures which are not regular if and only if the curvature at the puncture is 4π .

A polytope is the generalization of a polyhedron to any number of dimensions. The regular polyhedra are the Platonic solids: the tetrahedron, octahedron, cube, icosahedron, and dodecahedron. The hypercubes, hyperoctahedra, simplices, and regular polygons form four infinite fa milies of regular polytopes. Ludwig Schläfli proved that with the addition of five exceptional solids (the icosahedron and dodecahedron in 3 dimensions, and the 24-cell, 120-cell, and 600-cell in 4 dimensions) this list is complete. This paper provides an alternate proof to Schläfli's result, using Wythoff's construction and the theory of decorated Coxeter diagrams.