Slicing convex bodies—bounds for slice area in terms of the body’s covariance

Hensley, Douglas

doi:10.1090/S0002-9939-1980-0572315-5

Slicing convex bodies—bounds for slice area in terms of the body’s covariance
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Proc. Amer. Math. Soc. 79 (1980), 619-625 Request permission

Abstract:

Let Q be a zero-symmetric convex set in ${{\mathbf {R}}^N}$ with volume 1 and covariance matrix $V^2 \mathrm {Id}_{N \times N}$. Let P be a K-dimensional vector subspace of ${{\mathbf {R}}^n},K < N$, and let $J = N - K$. Then there exist constants ${C_1}(J)$ and ${C_2}(J)$ such that \[ {V^{ - J}}{C_1}(J) \leqslant \mathrm {vol}_K(P \cap Q) \leqslant V^{-J}{C_2}(J).\] The lower bound has applications to Diophantine equations.

References

Leo Breiman, Probability, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1968. MR 0229267
Ju. S. Davidovič, B. I. Korenbljum, and B. I. Hacet, A certain property of logarithmically concave functions, Dokl. Akad. Nauk SSSR 185 (1969), 1215–1218 (Russian). MR 0241584
Howard Eves, Elementary matrix theory, Allyn and Bacon, Inc., Boston, Mass., 1966. MR 0199198
Wendell Fleming, Functions of several variables, 2nd ed., Undergraduate Texts in Mathematics, Springer-Verlag, New York-Heidelberg, 1977. MR 0422527, DOI 10.1007/978-1-4684-9461-7
F. R. Gantmacher, Matrizenrechnung. Teil I: Allgemeine Theorie, Hochschulbücher für Mathematik, Band 36, VEB Deutscher Verlag der Wissenschaften, Berlin, 1970 (German). Dritte Auflage; Aus dem Russischen übersetzt von Klaus Stengert. MR 0279108
Douglas Hensley, Slicing the cube in $\textbf {R}^{n}$ and probability (bounds for the measure of a central cube slice in $\textbf {R}^{n}$ by probability methods), Proc. Amer. Math. Soc. 73 (1979), no. 1, 95–100. MR 512066, DOI 10.1090/S0002-9939-1979-0512066-8
Marek Kanter, Unimodality and dominance for symmetric random vectors, Trans. Amer. Math. Soc. 229 (1977), 65–85. MR 445580, DOI 10.1090/S0002-9947-1977-0445580-7
Henryk Minc and Marvin Marcus, Introduction to linear algebra, The Macmillan Company, New York; Collier Macmillan Ltd., London, 1965. MR 0188221
András Prékopa, On logarithmic concave measures and functions, Acta Sci. Math. (Szeged) 34 (1973), 335–343. MR 404557
Yosef Rinott, On convexity of measures, Ann. Probability 4 (1976), no. 6, 1020–1026. MR 428540, DOI 10.1214/aop/1176995947
Jeffrey D. Vaaler, A geometric inequality with applications to linear forms, Pacific J. Math. 83 (1979), no. 2, 543–553. MR 557952, DOI 10.2140/pjm.1979.83.543
Jeffrey D. Vaaler, On linear forms and Diophantine approximation, Pacific J. Math. 90 (1980), no. 2, 475–482. MR 600645, DOI 10.2140/pjm.1980.90.475