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Lecture# Isoperimetric Inequality: Optimal Transport

Description

This lecture focuses on the isoperimetric inequality and its surprising application in optimal transport. The instructor explains how the Brignier theorem is used to prove the inequality, showing that the perimeter of a set is greater or equal to the perimeter of a ball with the same volume. The lecture delves into the properties of the optimal transport map, the convexity of functions, and the uniqueness of optimizers. The instructor also discusses quality cases, where sets other than a ball minimize the perimeter at a given area. The lecture concludes with a discussion on the deficit of a set and a refined theorem by Alessio Figalli. The concepts are illustrated with examples and proofs, providing insights into the geometric implications of the isoperimetric inequality.

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In mathematics, a real-valued function is called convex if the line segment between any two distinct points on the graph of the function lies above the graph between the two points. Equivalently, a function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. A twice-differentiable function of a single variable is convex if and only if its second derivative is nonnegative on its entire domain.

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In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points in the subset, the subset contains the whole line segment that joins them. Equivalently, a convex set or a convex region is a subset that intersects every line into a single line segment (possibly empty). For example, a solid cube is a convex set, but anything that is hollow or has an indent, for example, a crescent shape, is not convex. The boundary of a convex set is always a convex curve.

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