Recorded 07 February 2022. Bo'az Klartag of the Weizmann Institute of Science presents "Size of line intersections in high dimensional L_p-balls and product measures" at IPAM's Calculus of Variations in Probability and Geometry Workshop. Abstract: Suppose that K is an n-dimensional convex body of volume one. The isodiametric inequality states that K contains a line segment whose length has the order of magnitude of sqrt{n}. Does there exist a subset A of K of volume 1/2, such that for any line in R^n, its intersection with A is of measure much smaller than sqrt{n}? Joint work with D. Elboim.