Finite homogeneous metric spaces with special properties

Abstract: This talk is devoted to some recent results on finite homogeneous metric spaces obtained in joint papers with Prof. V.N. Berestovskii. Every finite homogeneous metric subspace of an Euclidean space represents the vertex set of a compact convex polytope with the isometry group that is transitive on the set of vertices, moreover, all these vertices lie on some sphere. Consequently, the study of such subsets is closely related to the theory of convex polytopes in Euclidean spaces.

The main subject of discussion is the classification of regular and semiregular polytopes in Euclidean spaces by whether or not their vertex sets have the normal homogeneity property or the Clifford - Wolf homogeneity property. The normal homogeneity and the Clifford - Wolf homogeneity describe more stronger properties than the homogeneity. Therefore, it is quite natural to check the presence of these properties for the vertex sets of regular and semiregular polytopes.

In the second part of the talk, we consider the $m$-point homogeneity property and the point homogeneity degree for finite metric spaces. Among main results, there is a classification of polyhedra with all edges of equal length and with 2-point homogeneous vertex sets.

The most recent results and still unsolved problems in this topic will also be discussed.

differential geometrymetric geometry

Audience: advanced learners

( video )

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Virtual seminar on geometry with symmetries

Series comments: Description: Research seminar in Lie group actions in Riemannian geometry.

The seminar meets every other Wednesday. To accommodate most time zones, the time rotates. The Zoom link is sent to the mailing list around 24 hours before each talk. To subscribe to the mailing list, send a message to geometrywithsymmetries@gmail.com requiring it.

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