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The Ultimate Frontier: An Optimality Construction for Homotopy Inference (Media Exposition) Attali, Dominique Dal Poz Kouřimská, Hana Fillmore, Christopher Ghosh, Ishika Lieutier, André Stephenson, Elizabeth Wintraecken, Mathijs Homotopy Inference Sets of positive reach In our companion paper "Tight bounds for the learning of homotopy à la Niyogi, Smale, and Weinberger for subsets of Euclidean spaces and of Riemannian manifolds" we gave optimal bounds (in terms of the two one-sided Hausdorff distances) on a sample P of an input shape 𝒮 (either manifold or general set with positive reach) such that one can infer the homotopy of 𝒮 from the union of balls with some radius centred at P, both in Euclidean space and in a Riemannian manifold of bounded curvature. The construction showing the optimality of the bounds is not straightforward. The purpose of this video is to visualize and thus elucidate said construction in the Euclidean setting. Schloss Dagstuhl – Leibniz-Zentrum für Informatik Dominique Attali and Hana Dal Poz Kouřimská and Christopher Fillmore and Ishika Ghosh and André Lieutier and Elizabeth Stephenson and Mathijs Wintraecken 2024 Is Part Of LIPIcs, Volume 293, 40th International Symposium on Computational Geometry (SoCG 2024) InProceedings Text doc-type:ResearchArticle publishedVersion application/pdf doi:10.4230/LIPIcs.SoCG.2024.87 urn:nbn:de:0030-drops-200325 https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SoCG.2024.87 eng https://creativecommons.org/licenses/by/4.0/legalcode