Lewis Marsh,David Beers
Lewis Marsh
The Euler characteristic transform (ECT) is a signature from topological data analysis (TDA) which summarises shapes embedded in Euclidean space. Compared with other TDA methods, the ECT is fast to compute and it is injective on a broad cla...
Orr Dunkelman,Zeev Geyzel,Chaya Keller et al.
Orr Dunkelman et al.
A major research area in discrete geometry is to consider the best way to partition the d-dimensional Euclidean space R d under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the...
Brittany Terese Fasy,Samuel Micka,David L Millman et al.
Brittany Terese Fasy et al.
The persistent homology transform, Betti function transform, and Euler characteristic transform represent a shape with a multiset of persistence diagrams, Betti functions, or Euler characteristic functions, respectively, parameterized by th...
Herbert Edelsbrunner,János Pach
Herbert Edelsbrunner
The Upper Bound Theorem for convex polytopes implies that the p-th Betti number of the Čech complex of any set of N points in R d and any radius satisfies β p = O ( N m ) , with m = min { p + 1 , ⌈ d / 2 ⌉ } . We c...
Yohai Reani,Omer Bobrowski
Yohai Reani
We study the k-th nearest neighbor distance function from a finite point-set in R d . We provide a Morse theoretic framework to analyze the sub-level set topology. In particular, we present a simple combinatorial-geometric characterizatio...
Ranita Biswas,Sebastiano Cultrera di Montesano,Ondřej Draganov et al.
Ranita Biswas et al.
Given a locally finite set A ⊆ R d and a coloring χ : A → { 0 , 1 , … , s } , we introduce the chromatic Delaunay mosaic of χ , which is a Delaunay mosaic in R d + s that represents how points of different ...
Ángel Javier Alonso,Michael Kerber
Ángel Javier Alonso
We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and firs...
Lech Duraj,Ross J Kang,Hoang La et al.
Lech Duraj et al.
Given a positive integer d, the class d-DIR is defined as all those intersection graphs formed from a finite collection of line segments in R 2 having at most d slopes. Since each slope induces an interval graph, it easily follows for eve...
Agelos Georgakopoulos
Agelos Georgakopoulos
Mohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and t...
Nóra Frankl,Panna Gehér,Arsenii Sagdeev et al.
Nóra Frankl et al.
We prove that for any ℓ p -norm in the plane with 1