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Explore the intriguing world of convex geometry and measure theory in this 25-minute talk from the Hausdorff Center for Mathematics. Delve into the properties of convex bodies and their relationships, starting with the fundamental concept that a convex body K contained within another convex body L implies a smaller surface area for K. Examine the supermodularity inequality for Minkowski sums of convex bodies A, B, and C using Lebesgue measure. Investigate weighted analogues of these properties by replacing Lebesgue measure with Borel measures, and learn about recent findings by G. Saracco and G. Stefani regarding monotonicity properties of measures with density. Analyze the supermodularity property for Radon measures and its equivalence to a variant of the monotonicity problem. Discover the conclusion that a Radon measure exhibiting supermodularity must be the Lebesgue measure. Finally, consider restricted versions of this problem to gain a deeper understanding of measure theory in the context of compact, convex sets.
