Randomized Isoperimetric Inequalities

Paouris, Grigoris; Pivovarov, Peter

doi:10.1007/978-1-4939-7005-6_13

Cited by 21 publications

(41 citation statements)

References 54 publications

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“…[17]. We only note that the importance of spindle convexity lies, at least partly, in the role intersections of congruent balls play in the study of, for example, the Kneser–Poulsen conjecture, diametrically complete bodies and randomized isoperimetric inequalities for more on this topic and references we suggest to see [2, 8, 11, 12, 17, 19].…”

Section: Introduction and Resultsmentioning

confidence: 99%

On Random Approximations by Generalized Disc‐polygons

2020

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For two convex discs K and L, we say that K is L-convex (Lángi et al., Aequationes Math. 85(1-2) (2013), 41-67) if it is equal to the intersection of all translates of L that contain K. In L-convexity, the set L plays a similar role as closed half-spaces do in the classical notion of convexity. We study the following probability model: Let K and L be C 2 + smooth convex discs such that K is L-convex. Select n independent and identically distributed uniform random points x 1 , . . . , x n from K, and consider the intersection K (n) of all translates of L that contain all of x 1 , . . . , x n . The set K (n) is a random L-convex polygon in K. We study the expectation of the number of vertices f 0 (K (n) ) and the missed area A(K \ K n ) as n tends to infinity. We consider two special cases of the model. In the first case, we assume that the maximum of the curvature of the boundary of L is strictly less than 1 and the minimum of the curvature of K is larger than 1. In this setting, the expected number of vertices and missed area behave in a similar way as in the classical convex case and in the r-spindle convex case (when L is a radius r circular disc), see (Fodor et al., Adv. in Appl. Probab. 46 (4) (2014), [899][900][901][902][903][904][905][906][907][908][909][910][911][912][913][914][915][916][917][918]. The other case we study is when K = L. This setting is special in the sense that an interesting phenomenon occurs: the expected number of vertices tends to a finite limit depending only on L. This was previously observed in the special case when L is a circle of radius r in Fodor et al. (Adv. in Appl. Probab. 46 (4) (2014), 899-918). We also determine the extrema of the limit of the expectation of the number of vertices of L (n) if L is a convex discs of constant width 1. The formulas we prove can be considered as generalizations of the corresponding r-spindle convex statements proved by Fodor et al. in (Adv. in Appl. Probab. 46(4) (2014), 899-918).

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Section: Introduction and Resultsmentioning

confidence: 99%

On Random Approximations by Generalized Disc‐polygons

2020

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“…For applications of (4.18), the reader is referred to [10,13,15,26,27,50,51,70]. A similar inequality on the sphere and the hyperbolic spaces was studied by Dann, Kim and Yaskin [14].…”

Section: )mentioning

confidence: 99%

Norm estimates for k-plane transforms and geometric inequalities

Rubin

2019

Advances in Mathematics

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The article is devoted to remarkable interrelation between the norm estimates for k-plane transforms in weighted and unweighted L p spaces and geometric integral inequalities for crosssections of measurable sets in R n . We also consider more general j-plane to k-plane transforms on affine Grassmannians and their compact modifications. The article contains a series of new integral-geometric inequalities with sharp constants, explicit equalities, conjectures, and open problems.

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“…The randomized forms of Urysohn's inequality (2) and (3) are just two examples of stochastic inequalities in

R^{n}

. Other fundamental inequalities like those of Brunn–Minkowski [24] and Blaschke–Santaló [38] also admit stochastic strengthenings [21, 35]. Moreover, centroid bodies and their

L_{p}

[29] and Orlicz extensions [30] also admit stochastic forms [32, 35].…”

Section: Introductionmentioning

confidence: 99%

“…Rather, we first prove a rearrangement inequality for

I (f_{1}, \dots, f_{N})

that reduces the problem to radially decreasing densities. This is similar to the route taken in the Euclidean setting [32, 35] but we use two‐point symmetrization, for example, [6, 14, 42], as in [23], rather than Steiner symmetrization. In

R^{n}

, a key tool behind stochastic isoperimetric inequalities is Steiner symmetrization and associated rearrangement inequalities [13, 20, 37].…”

Section: Introductionmentioning

confidence: 99%

Randomized Urysohn–type Inequalities

Hack

Pivovarov

2020

Mathematika

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As a natural analog of Urysohn's inequality in Euclidean space, Gao, Hug, and Schneider showed in 2003 that in spherical or hyperbolic space, the total measure of totally geodesic hypersurfaces meeting a given convex body K is minimized when K is a geodesic ball. We present a random extension of this result by taking K to be the convex hull of finitely many points drawn according to a probability distribution and by showing that the minimum is attained for uniform distributions on geodesic balls. As a corollary, we obtain a randomized Blaschke–Santaló inequality on the sphere.

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Randomized Isoperimetric Inequalities

Cited by 21 publications

References 54 publications

On Random Approximations by Generalized Disc‐polygons

On Random Approximations by Generalized Disc‐polygons

Norm estimates for k-plane transforms and geometric inequalities

Randomized Urysohn–type Inequalities

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