@inproceedings{6647,
  abstract     = {The Tverberg theorem is one of the cornerstones of discrete geometry. It states that, given a set X of at least (d+1)(r-1)+1 points in R^d, one can find a partition X=X_1 cup ... cup X_r of X, such that the convex hulls of the X_i, i=1,...,r, all share a common point. In this paper, we prove a strengthening of this theorem that guarantees a partition which, in addition to the above, has the property that the boundaries of full-dimensional convex hulls have pairwise nonempty intersections. Possible generalizations and algorithmic aspects are also discussed. As a concrete application, we show that any n points in the plane in general position span floor[n/3] vertex-disjoint triangles that are pairwise crossing, meaning that their boundaries have pairwise nonempty intersections; this number is clearly best possible. A previous result of Alvarez-Rebollar et al. guarantees floor[n/6] pairwise crossing triangles. Our result generalizes to a result about simplices in R^d,d >=2.},
  author       = {Fulek, Radoslav and Gärtner, Bernd and Kupavskii, Andrey and Valtr, Pavel and Wagner, Uli},
  booktitle    = {35th International Symposium on Computational Geometry},
  isbn         = {9783959771047},
  issn         = {1868-8969},
  location     = {Portland, OR, United States},
  pages        = {38:1--38:13},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{The crossing Tverberg theorem}},
  doi          = {10.4230/LIPICS.SOCG.2019.38},
  volume       = {129},
  year         = {2019},
}

@inproceedings{6648,
  abstract     = {Various kinds of data are routinely represented as discrete probability distributions. Examples include text documents summarized by histograms of word occurrences and images represented as histograms of oriented gradients. Viewing a discrete probability distribution as a point in the standard simplex of the appropriate dimension, we can understand collections of such objects in geometric and topological terms. Importantly, instead of using the standard Euclidean distance, we look into dissimilarity measures with information-theoretic justification, and we develop the theory
needed for applying topological data analysis in this setting. In doing so, we emphasize constructions that enable the usage of existing computational topology software in this context.},
  author       = {Edelsbrunner, Herbert and Virk, Ziga and Wagner, Hubert},
  booktitle    = {35th International Symposium on Computational Geometry},
  isbn         = {9783959771047},
  location     = {Portland, OR, United States},
  pages        = {31:1--31:14},
  publisher    = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik},
  title        = {{Topological data analysis in information space}},
  doi          = {10.4230/LIPICS.SOCG.2019.31},
  volume       = {129},
  year         = {2019},
}