The Brunn-Minkowski inequality

Gardner, Richard J.

doi:10.1090/s0273-0979-02-00941-2

Cited by 756 publications

(585 citation statements)

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“…We 4 The optimization for the inequalities of power means obtain not only a necessary and sufficient condition, but also an interesting sufficient condition such that inequality (1.6) holds. Note that the inequalities (1.6), (1.8), and (1.10) play some roles in the geometry of convex body (see, e.g., [3,7]). Our methods are, of late years, the approach of descending dimension and theory of majorization; and apply some techniques of mathematical analysis and permanents [12] in algebra.…”

Section: Definition 12mentioning

confidence: 99%

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The optimization for the inequalities of power means

Wen

Wang

2006

Journal of Inequalities and Applications

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Let M [t] n (a) be the tth power mean of a sequence a of positive real numbers, where a = (a 1 ,a 2 ,...,a n ),n ≥ 2, and α,λ ∈ R m ++ ,m ≥ 2, m j=1 λ j = 1,min{α} ≤ θ ≤ max {α}. In this paper, we will state the important background and meaning of the inequalityn (a); a necessary and sufficient condition and another interesting sufficient condition that the foregoing inequality holds are obtained; an open problem posed by Wang et al. in 2004 is solved and generalized; a rulable criterion of the semipositivity of homogeneous symmetrical polynomial is also obtained. Our methods used are the procedure of descending dimension and theory of majorization; and apply techniques of mathematical analysis and permanents in algebra.Copyright © 2006 J. Wen and W.-L. Wang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Symbols and introductionWe will use some symbols in the well-known monographs [1,5,13]:is a finite set, and it is not empty.Recall that the definitions of the tth power mean and Hardy mean of order r for a sequence a = (a 1 ,...,a n ) (n ≥ 2) are, respectively,, if 0< |t| < +∞,n (a) = n √ a 1 a 2 ··· a n , if t = 0, Hindawi Publishing Corporation

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Section: Definition 12mentioning

confidence: 99%

“…Let the measurable function on the measurable sets E and E 0 , Inequality (5.6) has important background in the geometry of convex body (see, e.g., [3,7]). Namely, for 0 < s ≤ 0.41904923394695076 ..., inequality (6.14) holds.…”

Section: The Sufficient Condition That Inequality (16) Holdsmentioning

confidence: 99%

The optimization for the inequalities of power means

Wen

Wang

2006

Journal of Inequalities and Applications

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“…This is essentially the content of Theorem 2 below. The following inequality has many uses in geometry, statistics, and analysis (see [16] for a proof, and [4] for more context, uses, and references). Note that it is stated with respect to a specific and not to all .…”

Section: Lemmamentioning

confidence: 99%

“…Using as the diameter of the truncated set, and observing that , and then applying Theorem 2, we find (III. 4) Noting that , we have…”

Section: Propositionmentioning

confidence: 99%

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An Inequality for Nearly Log-Concave Distributions with Applications to Learning

Caramanis

Mannor

2004

Learning Theory

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Abstract-We prove that given a nearly log-concave distribution, in any partition of the space to two well separated sets, the measure of the points that do not belong to these sets is large. We apply this isoperimetric inequality to derive lower bounds on the generalization error in learning. We further consider regression problems and show that if the inputs and outputs are sampled from a nearly log-concave distribution, the measure of points for which the prediction is wrong by more than 0 and less than 1 is (roughly) linear in 1 0 0 , as long as 0 is not too small, and 1 not too large. We also show that when the data are sampled from a nearly log-concave distribution, the margin cannot be large in a strong probabilistic sense.

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