Estimates of Loomis–Whitney type for intrinsic volumes

Campi, Stefano; Gronchi, Paolo

doi:10.1016/j.aam.2011.01.001

Cited by 16 publications

(21 citation statements)

References 27 publications

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“…, n− 2} and general K, it remains an open problem. (Though it is not mentioned in [7], inequality (5) for m = 1 confirms a conjecture of Betke and McMullen, namely, the case r = 1 and s = d − 1 of [3, Conjecture 3(a), p. 537].)…”

Section: Introductionsupporting

confidence: 62%

“…, K n are compact convex sets in R n . The notation V (K, i; L, n − i), for example, means that there are i copies of K and n − i copies of L. The quantity (7) V…”

Section: Preliminariesmentioning

confidence: 99%

“…Here V denotes volume, so that whereas (1) provides, in particular, an upper bound for the volume of a convex body in terms of volumes of its projections on coordinate hyperplanes, Meyer's inequality gives a lower bound in terms of the volumes of its sections by coordinate hyperplanes. In [7], variants of the Loomis-Whitney inequality were found in which an upper bound for V (K) is replaced by an upper bound for the intrinsic volume V m (K), for some m ∈ {1, . .…”

Section: Introductionmentioning

confidence: 99%

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Reverse and dual Loomis-Whitney-type inequalities

Campi

Gardner

Gronchi

2015

Trans. Amer. Math. Soc.

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Various results are proved giving lower bounds for the mth intrinsic volume V m (K), m = 1, . . . , n − 1, of a compact convex set K in R n , in terms of the mth intrinsic volumes of its projections on the coordinate hyperplanes (or its intersections with the coordinate hyperplanes). The bounds are sharp when m = 1 and m = n − 1. These are reverse (or dual, respectively) forms of the Loomis-Whitney inequality and versions of it that apply to intrinsic volumes. For the intrinsic volume V 1 (K), which corresponds to mean width, the inequality obtained confirms a conjecture of Betke and McMullen made in 1983.2010 Mathematics Subject Classification. Primary: 52A20, 52A40; secondary: 52A38.

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Section: Introductionsupporting

confidence: 62%

“…, K n are compact convex sets in R n . The notation V (K, i; L, n − i), for example, means that there are i copies of K and n − i copies of L. The quantity (7) V…”

Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Reverse and dual Loomis-Whitney-type inequalities

Campi

Gardner

Gronchi

2015

Trans. Amer. Math. Soc.

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“…Theorem 5.2 (Loomis-Whitney type inequality for intrinsic volumes [8]). Let X ⊆ R n be a compact set and let its intrinsic volumes be {V 0 (X), .…”

Section: Proof Of the Main Theorem: Part Bmentioning

confidence: 99%

Intrinsic Entropies of Log-Concave Distributions

Jog

Anantharam

2018

IEEE Trans. Inform. Theory

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The entropy of a random variable is well-known to equal the exponential growth rate of the volumes of its typical sets. In this paper, we show that for any log-concave random variable X, the sequence of the ⌊nθ⌋ th intrinsic volumes of the typical sets of X in dimensions n ≥ 1 grows exponentially with a well-defined rate. We denote this rate by h X (θ), and call it the θ th intrinsic entropy of X. We show that h X (θ) is a continuous function of θ over the range [0, 1], thereby providing a smooth interpolation between the values 0 and h(X) at the endpoints 0 and 1, respectively.

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“…When ν is replaced by the isotropic surface area measure of K, inequality (1.5) (without equality conditions) was proved by Giannopoulos and Papadimitrakis [20]. Further extensions and generalizations of the Loomis-Whitney inequality can be found in [5,6,[8][9][10]14].…”

Section: Introductionmentioning

confidence: 96%

The dual Loomis–Whitney inequality

Huang

2016

Bull. London Math. Soc.

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We establish a dual version of the Loomis–Whitney inequality for isotropic measures with complete equality conditions, where the sharp lower bound is given in terms of the volumes of hyperplane sections. For the special case of cross measures, we can drop the condition that the underlying body has centroid at the origin, yielding an independent proof of a result of Meyer's [‘A volume inequality concerning sections of convex sets’, Bull. London Math. Soc. 20 (1988) 151–155].

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Estimates of Loomis–Whitney type for intrinsic volumes

Cited by 16 publications

References 27 publications

Reverse and dual Loomis-Whitney-type inequalities

Reverse and dual Loomis-Whitney-type inequalities

Intrinsic Entropies of Log-Concave Distributions

The dual Loomis–Whitney inequality

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