@inproceedings{7952, abstract = {Isomanifolds are the generalization of isosurfaces to arbitrary dimension and codimension, i.e. manifolds defined as the zero set of some multivariate vector-valued smooth function f: ℝ^d → ℝ^(d-n). A natural (and efficient) way to approximate an isomanifold is to consider its Piecewise-Linear (PL) approximation based on a triangulation 𝒯 of the ambient space ℝ^d. In this paper, we give conditions under which the PL-approximation of an isomanifold is topologically equivalent to the isomanifold. The conditions are easy to satisfy in the sense that they can always be met by taking a sufficiently fine triangulation 𝒯. This contrasts with previous results on the triangulation of manifolds where, in arbitrary dimensions, delicate perturbations are needed to guarantee topological correctness, which leads to strong limitations in practice. We further give a bound on the Fréchet distance between the original isomanifold and its PL-approximation. Finally we show analogous results for the PL-approximation of an isomanifold with boundary. }, author = {Boissonnat, Jean-Daniel and Wintraecken, Mathijs}, booktitle = {36th International Symposium on Computational Geometry}, isbn = {978-3-95977-143-6}, issn = {1868-8969}, location = {Zürich, Switzerland}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The topological correctness of PL-approximations of isomanifolds}}, doi = {10.4230/LIPIcs.SoCG.2020.20}, volume = {164}, year = {2020}, } @inbook{74, abstract = {We study the Gromov waist in the sense of t-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov’s original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no t-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov’s pancake argument to produce some estimates of t-neighborhoods of (weighted) volume-critical submanifolds in the spirit of the waist theorems, including neighborhoods of algebraic manifolds in the complex projective space. In the appendix of this paper we provide for reader’s convenience a more detailed explanation of the Caffarelli theorem that we use to handle not necessarily radially symmetric Gaussian measures.}, author = {Akopyan, Arseniy and Karasev, Roman}, booktitle = {Geometric Aspects of Functional Analysis}, editor = {Klartag, Bo'az and Milman, Emanuel}, isbn = {9783030360191}, issn = {16179692}, pages = {1--27}, publisher = {Springer Nature}, title = {{Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures}}, doi = {10.1007/978-3-030-36020-7_1}, volume = {2256}, year = {2020}, } @article{7554, abstract = {Slicing a Voronoi tessellation in ${R}^n$ with a $k$-plane gives a $k$-dimensional weighted Voronoi tessellation, also known as a power diagram or Laguerre tessellation. Mapping every simplex of the dual weighted Delaunay mosaic to the radius of the smallest empty circumscribed sphere whose center lies in the $k$-plane gives a generalized discrete Morse function. Assuming the Voronoi tessellation is generated by a Poisson point process in ${R}^n$, we study the expected number of simplices in the $k$-dimensional weighted Delaunay mosaic as well as the expected number of intervals of the Morse function, both as functions of a radius threshold. As a by-product, we obtain a new proof for the expected number of connected components (clumps) in a line section of a circular Boolean model in ${R}^n$.}, author = {Edelsbrunner, Herbert and Nikitenko, Anton}, issn = {10957219}, journal = {Theory of Probability and its Applications}, number = {4}, pages = {595--614}, publisher = {SIAM}, title = {{Weighted Poisson–Delaunay mosaics}}, doi = {10.1137/S0040585X97T989726}, volume = {64}, year = {2020}, } @article{7666, abstract = {Generalizing the decomposition of a connected planar graph into a tree and a dual tree, we prove a combinatorial analog of the classic Helmholtz–Hodge decomposition of a smooth vector field. Specifically, we show that for every polyhedral complex, K, and every dimension, p, there is a partition of the set of p-cells into a maximal p-tree, a maximal p-cotree, and a collection of p-cells whose cardinality is the p-th reduced Betti number of K. Given an ordering of the p-cells, this tri-partition is unique, and it can be computed by a matrix reduction algorithm that also constructs canonical bases of cycle and boundary groups.}, author = {Edelsbrunner, Herbert and Ölsböck, Katharina}, issn = {14320444}, journal = {Discrete and Computational Geometry}, pages = {759--775}, publisher = {Springer Nature}, title = {{Tri-partitions and bases of an ordered complex}}, doi = {10.1007/s00454-020-00188-x}, volume = {64}, year = {2020}, } @article{7962, abstract = {A string graph is the intersection graph of a family of continuous arcs in the plane. The intersection graph of a family of plane convex sets is a string graph, but not all string graphs can be obtained in this way. We prove the following structure theorem conjectured by Janson and Uzzell: The vertex set of almost all string graphs on n vertices can be partitioned into five cliques such that some pair of them is not connected by any edge (n→∞). We also show that every graph with the above property is an intersection graph of plane convex sets. As a corollary, we obtain that almost all string graphs on n vertices are intersection graphs of plane convex sets.}, author = {Pach, János and Reed, Bruce and Yuditsky, Yelena}, issn = {14320444}, journal = {Discrete and Computational Geometry}, number = {4}, pages = {888--917}, publisher = {Springer Nature}, title = {{Almost all string graphs are intersection graphs of plane convex sets}}, doi = {10.1007/s00454-020-00213-z}, volume = {63}, year = {2020}, }