Mathematics

We show that compact Riemannian manifolds, regarded as metric spaces with their global geodesic distance, cannot contain a number of rigid structures such as (a) arbitrarily large regular simplices or (b) arbitrarily long sequences of points equidistant from pairs of points preceding them in the sequence. All of this provides evidence that Riemannian metric spaces admit what we term loose embeddings into finite-dimensional Euclidean spaces: continuous maps that preserve both equality as well as inequality. We also prove a local-to-global principle for Riemannian-metric-space loose embeddability: if every finite subspace thereof is loosely embeddable into a common R N , then the metric space as a whole is loosely embeddable into R N in a weakened sense.

We give another proof of Toyoda's theorem that describes 5-point subpaces in CAT(0) length spaces

We prove a C m Lusin approximation theorem for horizontal curves in the Heisenberg group. This states that every absolutely continuous horizontal curve whose horizontal velocity is m−1 times L 1 differentiable almost everywhere coincides with a C m horizontal curve except on a set of small measure. Conversely, we show that the result no longer holds if L 1 differentiability is replaced by approximate differentiability. This shows our result is optimal and highlights differences between the Heisenberg and Euclidean settings.

In 1933, Borsuk proposed the following problem: Can every bounded set in E n be divided into n+1 subsets of smaller diameters? This problem has been studied by many authors, and a lot of partial results have been discovered. In particular, Kahn and Kalai's counterexamples surprised the mathematical community in 1993. Nevertheless, the problem is still far away from being completely resolved. This paper presents a broad review on related subjects and, based on a novel reformulation, introduces a computer proof program to deal with this well-known problem.

In this paper we will develop an axiomatic foundation for the geometric study of straight edge, protractor, and compass constructions, which while being related to previous foundations, will be the first to have all axioms written and all proofs conducted in quantifier-free first order logic. All constructions within the system will be justified to be feasible by basic human faculties. No statement in the system will refer to infinitely many objects and one can posit an interpretation of the system which is in accordance to our free, creative process of geometric constructions. We are also able to capture analogous results to Euclid's work on non-planar geometry in Book XI of The Elements. This paper primarily builds on Suppes' paper Quantifier-Free Axioms for Constructive Affine Plane Geometry and draws from Beeson's article A Constructive Version of Tarski's Geometry. By further developing Suppes' work on parallel line segments, we are able to develop analogs to most theorems about parallel lines without assuming an equivalent to Euclid's Fifth Postulate which we deem as introducing non-feasible constructions. In A Constructive Version of Tarski's Geometry, Beeson defines the characteristics such a geometric foundation should have to called constructive. This work satisfies these characteristics. Additionally this work would be considered constructive as Suppes defined it.

In (Calc.Var.PDE 2018) Schultz generalized the work of Rajala and Sturm (Calc.Var.PDE 2014), proving that a weak non-branching condition holds in the more general setting of very strict CD spaces. Anyway, similar to what happens for the strong CD condition, the very strict CD condition seems not to be stable with respect to the measured Gromov Hausdorff convergence. In this article I prove a stability result for the very strict CD condition, assuming some metric requirements on the converging sequence and on the limit space. The proof relies on the notions of \textit{consistent geodesic flow} and \textit{consistent plan selection}, which allow to treat separately the static and the dynamic part of a Wasserstein geodesic. As an application, I prove that the metric measure space R N equipped with a crystalline norm and with the Lebesgue measure satisfies the very strict CD(0,?? condition.

We prove a Johns theorem for simplices in R d with positive dilation factor d+2 , which improves the previously known d 2 upper bound. This bound is tight in view of the d lower bound. Furthermore, we give an example that d isn't the optimal lower bound when d=2 . Our results answered both questions regarding Johns theorem for simplices with positive dilation raised by \cite{leme2020costly}.

We explore toroidal analogues of the Maxwell-Cremona correspondence. Erickson and Lin [arXiv:2003.10057] showed the following correspondence for geodesic torus graphs G : a positive equilibrium stress for G , an orthogonal embedding of its dual graph G ∗ , and vertex weights such that G is the intrinsic weighted Delaunay graph of its vertices. We extend their results to equilibrium stresses that are not necessarily positive, which correspond to orthogonal drawings of G ∗ that are not necessarily embeddings. We also give a correspondence between equilibrium stresses and parallel drawings of the dual.

In this paper, we prove a Prékopa-Leindler type inequality of the L p Brunn-Minkowski inequality. It extends an inequality proved by Das Gupta [8] and Klartag [16], and thus recovers the Prékopa-Leindler inequality. In addition, we prove a functional L p Minkowski inequality.

We give an alternative proof for discrete Brunn-Minkowski type inequalities, recently obtained by Halikias, Klartag and the author. This proof also implies somewhat stronger weighted versions of these inequalities. Our approach generalizes ideas of Gozlan, Roberto, Samson and Tetali from the theory of measure transportation and provides new displacement convexity of entropy type inequalities for the lattice point enumerator.