Shakhar Smorodinsky’s Solution to a Radon-Type Problem

A brief update: Since Friday June 13 Israel has been engaged in a direct war with Iran. This follows two major missiles attacks of Iran against Israel in April and October 2024, as well as Iran’s central role in the war since October 7 (and earlier).  Israel’s goals are to strike Iran’s nuclear program and its ballistic missile capabilities. These days, we are under missile attacks from Iran and spend time in shelters; planned academic events have been canceled or postponed, and academic teaching is now mainly on Zoom.

Shakhar Smorodinsky settled an old problem, on the spot!

I have much to report about recent academic events, both in Israel and abroad, but let me highlight one lovely development. At Micha Sharir’s birthday meeting, I mentioned an old Radon-type problem that I raised in the 1970s. The problem involves a Radon-type theorem where the usual assertion — “the convex hulls of the two parts intersect” — is replaced by: “every union of k convex sets containing the first part intersects every union of r convex sets containing the second part.” Shakhar Smorodinsky used a VC-dimension argument to solve the problem on the spot, and following further discussion, he and Noga Alon made additional progress (which, in turn, raises even more questions). This now looks to me like a very promising direction in discrete geometry, and I hope to share more about it in a future post.

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The problem that Shakhar solved was one I raised in the 1970s. (I may have posted it in one of the Oberwolfach problem collections in the 1980s.) It was Problem 7 in my lecture on Micha Sharir Day (slides are here). It appeared in Section 1.7 of the 2017 short paper discussed in this post, and later in my paper with Imre Bárány on Helly type problems.

Two pictures from Mehtaab S. Sawhney’s Erdős lecture.

From the first Erdos lecture by Mehtaab S. Sawhney.  On the blackboard, you can see the important “bracket polynomials”. To the right, a picture of Erdős. In the top-right corner, there is a digital thermometer. About 15–20 years ago, following complaints about the temperature in the room, the HUJI maintenance staff installed a large digital thermometer (showing 22.9°C in the picture). The thermometer caused some confusion, as speakers often mistook it for a clock or a timer. To address this, a few years later, a large digital clock (showing 3:12:30 in the picture) was installed.