Families of similar simplices inscribed in most smoothly embedded spheres
Jason Cantarella, Elizabeth Denne, and John McCleary

TL;DR
This paper studies families of similar simplices inscribed in smoothly embedded spheres, revealing dense subsets with all poses and connecting inscribed simplices to classical geometric problems.
Contribution
It generalizes classical inscribed triangle results to higher dimensions using configuration space techniques and explores the topology of inscribed simplices in embedded spheres.
Findings
Dense families of spheres with all simplex poses inscribed
Top homology class maps to the pose group $O(k)$
High-dimensional generalization of inscribed triangle results
Abstract
Let denote a non-degenerate -simplex in . The set of simplices in similar to is diffeomorphic to , where the factor in is a matrix called the {\em pose}. Among -spheres smoothly embedded in and isotopic to the identity, there is a dense family of spheres, for which the subset of of simplices inscribed in each embedded sphere contains a similar simplex of every pose . Further, the intersection of with the configuration space of distinct points on an embedded sphere is a manifold whose top homology class maps to the top class in via the pose map. This gives a high dimensional generalization of classical results on inscribing families of triangles in plane curves. We use techniques…
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Taxonomy
TopicsTopological and Geometric Data Analysis · Computational Geometry and Mesh Generation · Point processes and geometric inequalities
