Cube Slicing in R n

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. This content downloaded from 128.235.251.160 on TueABSTRACT. We prove that every (n -1)-dimensional section of the unit cube in R' has volume at most I2 This upper bound is clearly best possible.

Central slices of the regular simplex

“…This upper bound was proved by Ball [1], [2] who also considered the case for a general k-dimensional subspace H. For the regular simplex, it follows from results of Ball [3] that the k-dimensional slices with largest volume are exactly the k-dimensional faces. Results of this type provide estimates for densities of sums of i.i.d, random variables so perhaps a more interesting problem for the simplex is that of the central sections, as this will involve random variables with mean zero.…”

“…The proof of the theorem will divide into two parts, the first for the case lad I --< ~22 for each 1 _< j _< n + 1 and the second for the case lajl > for some 1 _< j _< n + 1. This is analogous to the proof for the cube in [1]. We use the same approach for the first part and apply an easier estimate than was required in [1], but for the second part, the easy part for the cube, we need a different method.…”

Solution of the propeller conjecture in R 3

“…In (40) replace θ 2 jr with the left side of (42) for (j, r) ∈ {(1, 4), (2,4), (3,4)} to get F (P ) = 3 θ Since the spherical propeller partition, i.e., the partition of S 2 corresponding to the intersection of the propeller partition of R 3 with S 2 , satisfies F = (9/4)π 2 , we may bound (43) in terms of M = max{θ 12 , θ 13 , θ 23 }. In particular, we get F (P ) 14M 2 , which is less than (9/4)π 2 if M < π…”

On Weak Tail Domination of Random Vectors

“…where to get the second inequality we used Ball's upper bound on cube sections [1]. Hence (3) holds also for t ∈ [0, 3].…”