Geometry Seminar

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On the heterogeneity in metric spaces

Speaker: Marc van Kreveld, Utrecht University

Location: Online

Date: Tuesday, October 6, 2026, 2 p.m.

Synopsis:

Heterogeneity is a property of sets that shows how varied or different its elements are. We define full heterogeneity in a metric space and study the maximum size of fully heterogeneous sets. A set is fully heterogeneous if each pair of elements is as distant as the maximum possible distance between any pair, up to a constant factor. We study bounded metric spaces based on geometry, embeddings of graphs, and graphs themselves.

In the geometric cases, we study measures like Hausdorff distance, Frechet distance, and area of symmetric difference between objects in a bounded region.

In the embedding cases, we study planar embeddings of trees and planar graphs, and use the number of swaps in the rotation system as the metric.

In the graph cases, we use the number of insertions and deletions of leaves or edges as the metric.

In most cases, we show (almost) "tight" lower and upper bounds on the maximum size of fully heterogeneous sets, with bounds like \(2^{O(n)}\) and \(2^{\Omega(n)}\).

Joint work with Fabian Klute.

Notes:

Only on Zoom.  Please contact Boris Aronov to be put on the email announcement list and obtain Zoom details.