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January 2022
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Hugo Lavenant
Monday 20 May 2019
Hugo Lavenant (Université Paris-Sud)
Isoperimetric profile in a domain with boundary and its convex relaxation.
Abstract : In an (a priori non convex) domain, we investigate the isoperimetric profile, i.e. we look for shapes of a given volume with the smallest perimeter that can be enclosed in the domain. Most of the existing literature consider the relative perimeter, while in our case the part of the perimeter located on the boundary of the domain counts. This investigation originated from issues in political redistricting, as a tool to characterize the "twistedness" of domains. For domains in 2d without necks, a geometric characterization of the minimizers can be found, generalizing the one for Cheeger sets. On the other hand, for arbitrary domains, we show that the convex relaxation of the profile still leads to interesting and nontrivial features, while being amenable to numerical computations.
This is a joint work with Daryl DeFord, Zachary Schutzman, Justin Solomon and Giorgio Saracco.