Volume of intersections and sections of the unit ball of ℓⁿ_{𝑝}

Schmuckenschläger, Michael

doi:10.1090/S0002-9939-98-04179-3

Volume of intersections and sections of the unit ball of $\ell ^n_p$
HTML articles powered by AMS MathViewer

by Michael Schmuckenschläger PDF

Proc. Amer. Math. Soc. 126 (1998), 1527-1530 Request permission

Abstract:

An asymptotic formula for the volume of the intersection of a suitable multiple of the unit ball of $\ell _p^n$ and the cube $[-1/2,1/2]^n$ will be proved. We also show that the isotropic constant of the unit ball of $\ell _n^p, 1\le p\le 2$, is bounded by $1/\sqrt {12}$.

References

Keith Ball, Logarithmically concave functions and sections of convex sets in $\textbf {R}^n$, Studia Math. 88 (1988), no. 1, 69–84. MR 932007, DOI 10.4064/sm-88-1-69-84
Keith Ball, Normed spaces with a weak-Gordon-Lewis property, Functional analysis (Austin, TX, 1987/1989) Lecture Notes in Math., vol. 1470, Springer, Berlin, 1991, pp. 36–47. MR 1126735, DOI 10.1007/BFb0090210
Kai Lai Chung, A course in probability theory, 2nd ed., Probability and Mathematical Statistics, Vol. 21, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1974. MR 0346858
Mathieu Meyer and Alain Pajor, Sections of the unit ball of $L^n_p$, J. Funct. Anal. 80 (1988), no. 1, 109–123. MR 960226, DOI 10.1016/0022-1236(88)90068-7
V. D. Milman and A. Pajor, Isotropic position and inertia ellipsoids and zonoids of the unit ball of a normed $n$-dimensional space, Geometric aspects of functional analysis (1987–88), Lecture Notes in Math., vol. 1376, Springer, Berlin, 1989, pp. 64–104. MR 1008717, DOI 10.1007/BFb0090049
G. Schechtman and M. Schmuckenschläger, Another remark on the volume of the intersection of two $L^n_p$ balls, Geometric aspects of functional analysis (1989–90), Lecture Notes in Math., vol. 1469, Springer, Berlin, 1991, pp. 174–178. MR 1122622, DOI 10.1007/BFb0089224