An Elementary Introduction to Modern Convex Geometry by Ball K.

By Ball K.

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In some cases where the exact solutions are known, the two properties above are a good deal easier to prove than the solutions: and in a great many situations, an exact isoperimetric inequality is not known, but the two properties are. The formal similarity between property 2 and Bernstein’s inequality of the last lecture is readily apparent. There are ways to make this similarity much more than merely formal: there are deviation inequalities that have implications for Lipschitz functions and imply Bernstein’s inequality, but we shall not go into them here.

The difference between Pr´ekopa–Leindler and H¨ older is that, in the former, the value m(z) may be much larger since it is a supremum over many pairs (x, y) satisfying z = (1 − λ)x + λy rather than just the pair (z, z). Though it generalises the Brunn–Minkowski inequality, the Pr´ekopa–Leindler inequality is a good deal simpler to prove, once properly formulated. The argument we shall use seems to have appeared first in [Brascamp and Lieb 1976b]. The crucial point is that the passage from sets to functions allows us to prove the inequality by induction on the dimension, using only the one-dimensional case.

J. Brascamp and E. H. Lieb, “On extensions of the Brunn–Minkowski and Pr´ekopa–Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation”, J. Funct. Anal. 22 (1976), 366–389. [Brøndsted 1983] A. Brøndsted, An introduction to convex polytopes, Graduate Texts in Math. 90, Springer, New York, 1983. 54 KEITH BALL [Busemann 1949] H. Busemann, “A theorem on convex bodies of the Brunn–Minkowski type”, Proc. Nat. Acad. Sci. USA 35 (1949), 27–31.

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