Blaschke-Santaló inequalities

Lutwak, Erwin; Zhang, Gaoyong

doi:10.4310/jdg/1214460036

Cited by 261 publications

(173 citation statements)

References 13 publications

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“…The sets Z p (µ K ) for p ≥ 1, when µ K is the uniform distribution on the convex body K, are called the L p -centroid bodies of K. They were introduced (under a different normalization) in [18]; their properties were also investigated in [23].…”

Section: P -Centroid Bodies and Related Setsmentioning

confidence: 99%

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On the infimum convolution inequality

Latała

Wojtaszczyk

2008

Studia Math.

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Abstract. We study the infimum convolution inequalities. Such an inequality was first introduced by B. Maurey to give the optimal concentration of measure behaviour for the product exponential measure. We show how IC inequalities are tied to concentration and study the optimal cost functions for an arbitrary probability measure µ. In particular, we prove an optimal IC inequality for product log-concave measures and for uniform measures on the n p balls. Such an optimal inequality implies, for a given measure, the central limit theorem of Klartag and the tail estimates of Paouris.

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Section: P -Centroid Bodies and Related Setsmentioning

confidence: 99%

“…We say that a probability measure µ on R n has comparable weak and strong moments with constant γ (CWSM(γ) for short) if (18) holds for any norm · on R n . Conjecture 3.…”

Section: Now Integrating By Parts Givesmentioning

confidence: 99%

On the infimum convolution inequality

Latała

Wojtaszczyk

2008

Studia Math.

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“…Lutwak [103] wrote an excellent survey. For still more recent progress, the reader can do no better than consult the work of Lutwak, Yang, and Zhang, for example, [109] and [111].…”

Section: S(k L ·) = S(k ·) + S(l ·)mentioning

confidence: 99%

The Brunn-Minkowski inequality

Gardner

2002

Bull. Amer. Math. Soc.

732

561

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Abstract. In 1978, Osserman [124] wrote an extensive survey on the isoperimetric inequality. The Brunn-Minkowski inequality can be proved in a page, yet quickly yields the classical isoperimetric inequality for important classes of subsets of R n , and deserves to be better known. This guide explains the relationship between the Brunn-Minkowski inequality and other inequalities in geometry and analysis, and some applications.

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“…Definition 2.1 (see [11]). For each compact star-shaped K ⊂ R n about the origin and the real number p ≥ 1, the L p -centroid body Γ p K of K is the origin-symmetric body whose support function is defined by…”

Section: Preliminariesmentioning

confidence: 99%

EXTREMUM PROPERTIES OF DUAL L_p-CENTROID BODY AND L_p-JOHN ELLIPSOID

Ma¹

2012

Bulletin of the Korean Mathematical Society

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Abstract. For 0 < p ≤ ∞ and a convex body K in R n , Lutwak, Yang and Zhang defined the concept of dual Lp-centroid body Γ −p K and LpJohn ellipsoid EpK. In this paper, we prove the following two results:(i) For any origin-symmetric convex body K, there exist an ellipsoid E and a parallelotope P such that for 1 ≤ p ≤ 2 and 0 < q ≤ ∞,For 2 ≤ p ≤ ∞ and 0 < q ≤ ∞,(ii) For any convex body K whose John point is at the origin, there exists a simplex T such that for 1 ≤ p ≤ ∞ and 0 < q ≤ ∞,p EqT and V (K) = V (T ).

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Blaschke-Santaló inequalities

Cited by 261 publications

References 13 publications

On the infimum convolution inequality

On the infimum convolution inequality

The Brunn-Minkowski inequality

EXTREMUM PROPERTIES OF DUAL L_p-CENTROID BODY AND L_p-JOHN ELLIPSOID

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Blaschke-Santaló inequalities

Cited by 261 publications

References 13 publications

On the infimum convolution inequality

On the infimum convolution inequality

The Brunn-Minkowski inequality

EXTREMUM PROPERTIES OF DUAL Lp-CENTROID BODY AND Lp-JOHN ELLIPSOID

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EXTREMUM PROPERTIES OF DUAL L_p-CENTROID BODY AND L_p-JOHN ELLIPSOID