Topology '90 by Walter D. Neumann, B. N. Apanasov, B. N. Apanasov, Alan W.

By Walter D. Neumann, B. N. Apanasov, B. N. Apanasov, Alan W. Reid, Ohio State University International Mathematical Research Institute

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This inequality is satisfied by k of the order of nM 2 γ2 . 3 guarantees the existence of spherical slices of K of large dimension, provided the average θ dσ M= S n−1 is not too small. Notice that we certainly have M ≤ 1 since x ≤ |x| for all x. 3 we need to get a lower estimate for M of the order of √ log n √ . n AN ELEMENTARY INTRODUCTION TO MODERN CONVEX GEOMETRY 51 This is where we must use the fact that B2n is the maximal volume ellipsoid in √ √ K. We saw in Lecture 3 that in this situation K ⊂ nB2n , so x ≥ |x|/ n for all x, and hence 1 M≥√ .

Kaˇsin, “The widths of certain finite-dimensional sets and classes of smooth functions”, Izv. Akad. Nauk SSSR Ser. Mat. 41:2 (1977), 334–351, 478. In Russian. [Lieb 1990] E. H. Lieb, “Gaussian kernels have only Gaussian maximizers”, Invent. Math. 102 (1990), 179–208. [Lıusternik 1935] L. A. Lıusternik, “Die Brunn–Minkowskische Ungleichung f¨ ur beliebige messbare Mengen”, C. R. Acad. Sci. URSS 8 (1935), 55–58. [Maurey 1991] B. Maurey, “Some deviation inequalities”, Geom. Funct. Anal. 1:2 (1991), 188–197.

If x ∈ Ar and y ∈ At , the point (1 − λ)x + λy belongs to As : to see this, join the points (r, x) and (t, y) in Rn and observe that the resulting line segment crosses As at (s, (1 − λ)x + λy). So As includes a new set (1 − λ)Ar + λAt := {(1 − λ)x + λy : x ∈ Ar , y ∈ At } . (t, y) (s, x) Ar As At Figure 21. The section As contains the weighted average of Ar and At . ) Brunn’s Theorem says that the volumes of the three sets Ar , As , and At in Rn−1 satisfy 1/(n−1) vol (As ) 1/(n−1) ≥ (1 − λ) vol (Ar ) 1/(n−1) + λ vol (At ) .

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