Ball, Haagerup, and Distribution Functions

Nazarov, Fëdor; Podkorytov, Anatolii

doi:10.1007/978-3-0348-8378-8_21

Cited by 33 publications

(46 citation statements)

References 4 publications

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“…To prove Theorem , we use a modification of the method of Nazarov and Podkorytov in who simplified the proof of Ball's integral inequality in the case

m = 1

.…”

Section: Introduction and Resultsmentioning

confidence: 99%

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Volume estimates for sections of certain convex bodies

Brzezinski

2013

Mathematische Nachrichten

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Key words Regular simplex, cube slicing, probabilistic methods, volume of sections, Bessel functions MSC (2010) 52A20, 52A21, 46B07, 46B09, 33C10 This paper deals with volume estimates for hyperplane sections of the simplex and for m-codimensional sections of powers of m-dimensional Euclidean balls. In the first part we consider sections through the centroid of the n-dimensional regular simplex. We state a volume formula and give a lower bound for the volume of sections through the centroid. In the second part we study the extremal volumes of m-codimensional sectionsx ∈ B ∞,2 : n j=1 a j x j = 0 "perpendicular" to a ∈ S n−1 ⊆ R n of unit balls B ∞,2 in the space l n ∞ (l m 2 ) for all m, n ∈ N. We give volume formulas and use them to show that the normal vector (1, 0, . . . , 0) yields the minimal volume. Furthermore we give an upper bound for the m(n − 1)-dimensional volumes for natural numbers m ≥ 3. This bound is asymptotically attained for the normal vector 1 √ n , . . . , 1 √ n as n → ∞.

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“…To prove Theorem , we use a modification of the method of Nazarov and Podkorytov in who simplified the proof of Ball's integral inequality in the case

m = 1

.…”

Section: Introduction and Resultsmentioning

confidence: 99%

“…As noted above, we follow the ideas of Nazarov and Podkorytov, who used the following lemma [, pp. 247–267].…”

Section: Sections In the Generalized Cubementioning

confidence: 99%

Volume estimates for sections of certain convex bodies

Brzezinski

2013

Mathematische Nachrichten

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“…(In our bound on F (P ), we use that one Θ ijk must be larger than π/3 by the pigeonhole principle applied to Θ 123 + Θ 231 + Θ 312 − π = Area S 2 (T 4 (34) gives the following improvement to Proposition 3.1(e). This improvement will be important in our numerical calculations in Section 4 and in the proof of Theorem 1.1, since we eventually require a bound on the derivative of θ ij / sin θ ij .…”

Section: Proposition 32 Suppose That the Spherical Trianglementioning

confidence: 99%

Solution of the propeller conjecture in R ³

Heilman

Jagannath

2012

Proceedings of the Forty-Fourth Annual ACM Symposium on Theory of Computing

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Abstract. It is shown that every measurable partition {A1, . . . , A k } of R 3 satisfiesLet {P1, P2, P3} be the partition of R 2 into 120• sectors centered at the origin. The bound (1) is sharp, with equality holding if Ai = Pi × R for i ∈ {1, 2, 3} and Ai = ∅ for i ∈ {4, . . . , k}. This settles positively the 3-dimensional Propeller Conjecture of Khot and Naor (FOCS 2008). The proof of (1) reduces the problem to a finite set of numerical inequalities which are then verified with full rigor in a computer-assisted fashion. The main consequence (and motivation) of (1) is complexity-theoretic: the Unique Games hardness threshold of the Kernel Clustering problem with 4 × 4 centered and spherical hypothesis matrix equals . Figure 1. The partition of R 3 that maximizes the sum of squared lengths of Gaussian moments is a "propeller": three planar 120 • sectors multiplied by an orthogonal line, with the rest of the partition elements being empty.

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“…To prove by probabilistic methods that every

(n - 1)

‐dimensional section of the unit cube in

R^{n}

has volume at most

\sqrt{2}

, K. Ball [1] made essential use of the inequality

\frac{1}{π} \int_{- \infty}^{\infty} {((), \frac{\sin^{2} t}{t^{2}})}^{p} d t = \int_{- \infty}^{\infty} {((), \frac{\sin^{2} false(π t false)}{{false(π t false)}^{2}})}^{p} d t ⩽ \frac{\sqrt{2}}{\sqrt{p}}, p ⩾ 1,

in which equality holds if and only if

p = 1

. Later, this inequality was proven by using different methods (see [2, 4]).…”

Section: Introductionmentioning

confidence: 99%

An asymptotically sharp form of Ball's integral inequality by probability methods

Spektor

2021

Bull. London Math. Soc.

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Ball, Haagerup, and Distribution Functions

Cited by 33 publications

References 4 publications

Volume estimates for sections of certain convex bodies

Volume estimates for sections of certain convex bodies

Solution of the propeller conjecture in R ³

An asymptotically sharp form of Ball's integral inequality by probability methods

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Ball, Haagerup, and Distribution Functions

Cited by 33 publications

References 4 publications

Volume estimates for sections of certain convex bodies

Volume estimates for sections of certain convex bodies

Solution of the propeller conjecture in R 3

An asymptotically sharp form of Ball's integral inequality by probability methods

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Product

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Solution of the propeller conjecture in R ³