On the multivariable version of Ball’s slicing cube theorem. (English) Zbl 1077.52004

Milman, V. D. (ed.) et al., Geometric aspects of functional analysis. Papers from the Israel seminar (GAFA) 2002–2003. Berlin: Springer (ISBN 3-540-22360-6/pbk). Lecture Notes in Mathematics 1850, 117-121 (2004).
The author proves the following far-reaching generalization of K. Ball’s cube slicing theorem [“Volumes of sections of cubes and related problems”, Lect. Notes Math. 1376, 251–260 (1989; Zbl 0674.46008)]. Let \(V_j\) (\(j=1,2,\dots ,m\)) be compact subsets of \(\mathbb R^n\) (\(n\)-dimensional Euclidean space) each of Lebesgue measure one. Let \(H_C\) be the subspace of \(\mathbb{R}^{nm}\) defined by \(H_C:= \{(x_1,x_2, \dots,x_m) \in \mathbb R^{nm} : \sum_{j=1}^m c_{ij}x_j = 0 \text{ for } i = 1,2,\dots,k\}\) where \(C = (c_{ij})\) is an arbitrary \(k\times m\) real matrix and the \(x_j\)’s are in \(\mathbb R^n\). Then \(\text{meas}_H\{H_C\cap(V_1\times V_2 \times \cdots V_m)\} \leq \exp(kn/2)\). The proof uses Ball’s argument and F. Barthe’s multidimensional version of the Brascamp-Lieb inequality [Invent. Math. 134, 335–361 (1998; Zbl 0901.26010)]. Evidently one can translate the \(V_j\)’s and so the result applies to flats \(H_C\) as well as subspaces.
For the entire collection see [Zbl 1052.46001].


52A20 Convex sets in \(n\) dimensions (including convex hypersurfaces)
52A38 Length, area, volume and convex sets (aspects of convex geometry)