Geometry Seminar

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Helly-Type Theorems for Splitting Point Sets

Speaker: Lidor Portal, Ben Gurion U

Location: Warren Weaver Hall 1314 and on Zoom

Date: Tuesday, January 20, 2026, 6 p.m.

Synopsis:

Let \(0 < \alpha \leq 1/2\). We say that a finite point set \(P\) in \(\mathbb{R}^d\) is \(\alpha\)-split by a hyperplane \(h\) if each of the closed half-spaces determined by \(h\), contains at least \(\alpha |P|\) of the points of \(P\). We further say \(P\) is \(\alpha\)-split by a \(k\)-dimensional flat \(\tau\) if \(P\) is \(\alpha\)-split by any hyperplane through \(\tau\). In the standard notation (which coincides with Tukey depth for \(k=0 \)), the \(k\)-flat \(\tau\) has depth \(\alpha\) with respect to \(P\).

We establish interesting Helly-type theorems for splitting families of finite point sets in \(\mathbb{R}^d\). Unlike the classical sufficient Helly-type criteria for transversals to families of compact convex sets, which exist only for points and hyperplanes, our results extend to splitting families of point sets by collections of \(k\)-flats of arbitrary dimensionality \(0 \leq k \leq d-1\).

Joint work with Natan Rubin.

Notes:

In person and on Zoom.  Plz contact Boris Aronov for if you are not NYU-affiliated and want to attend in person, and/or to be put on the mailing list and get the Zoom information.