Dual mixed volumes and the slicing problem

By Emanuel Milman. 2005

53 citations recorded by the source corpus.

We develop a technique using dual mixed-volumes to study the isotropic constants of some classes of spaces. In particular, we recover, strengthen and generalize results of Ball and Junge concerning the isotropic constants of subspaces and quotients of Lp and related spaces. An extension of these results to negative values of p is also obtained, using generalized intersection-bodies. In particular, we show that the isotropic constant of a convex body which is contained in an intersection-body is bounded (up to a constant) by the ratio between the latter's mean-radius and the former's volume-radius. We also show how type or cotype 2 may be used to easily prove inequalities on any isotropic measure.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29227cc3778c386093672c2b30b26b108d79c3c7

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In Applied Mathematics

In Mathematical Physics

Citations in the corpus

On the hyperplane conjecture for random convex sets (2006, unknown): “M G ] 1 8”
The dual Orlicz–Brunn–Minkowski theory ☆ (2014, background): “ Minkowski geometry, the local theory of Banach spaces, geometric tomography, and stereology; see [13] and the references given there. The literature is large and continues to grow. See, for example, [1, 6, 10, 14, 15, 19, 24, 32]. 2010 Mathematics Subject Classiﬁcation. Primary: 52A20, 52A30; secon”
On convex perturbations with a bounded isotropic constant (2006, background): “The hyperplane conjecture was verified for several large classes of convex sets: Unconditional convex bodies [4, 21], zonoids, duals to zonoids, [2] (see also [ 20 ]), bodies with a bounded outer volume ratio [21], random bodies [18], unit balls of Schatten norms [19], and others (e.g., [15]).”
Dual Orlicz–Brunn–Minkowski theory: Dual Orlicz Lϕ affine and geominimal surface areas (2014, background): “The dual (Lp) BrunnMinkowski theory for star bodies received considerable attention, see [2, 4, 8, 12, 13, 14, 27, 33, 53] among others.”