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 ) .