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Slicing convex sets and measures by a hyperplane
[article]

2010
*
arXiv
*
pre-print

,α_d ∈ [0, 1], we give

arXiv:1010.6279v1
fatcat:zkf4m3epmbh4jequgjdmcotroq
*a*sufficient condition for existence*and*uniqueness of an (oriented)halfspace H with Vol(H ∩ K_i)= α_i VolK_i for every i. The result is extended from*convex*bodies to*measures*. ... We generalize the ham sandwich theorem for the case of well separated*measures*. Given*convex*bodies K_1,...,K_d in R_d*and*numbers α_1,... ... The first author was partially supported*by*Hungarian National Foundation Grants T 60427*and*NK 62321. ...##
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Slicing Convex Sets and Measures by a Hyperplane

2007
*
Discrete & Computational Geometry
*

*A*similar thing happens with d-pointed

*sets*

*and*

*hyperplanes*. What happens if we consider

*convex*bodies instead of points? I. Bárány ( ) Rényi ... Keywords

*Convex*bodies · Well separated families · Sections of

*convex*

*sets*

*and*

*measures*Transversal Spheres

*A*well known result in elementary geometry states that there is

*a*unique sphere which contains ... Acknowledgements The first author was partially supported

*by*Hungarian National Foundation Grants T 60427

*and*NK 62321. ...

##
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Entropy and the hyperplane conjecture in convex geometry

2010
*
2010 IEEE International Symposium on Information Theory
*

The

doi:10.1109/isit.2010.5513619
dblp:conf/isit/BobkovM10
fatcat:k6uvkmqxwraytkastxf4yy2h4u
*hyperplane*conjecture is*a*major unsolved problem in high-dimensional*convex*geometry that has attracted much attention in the geometric*and*functional analysis literature. ... It asserts that there exists*a*universal constant c such that for any*convex**set*K of unit volume in any dimension, there exists*a**hyperplane*H passing through its centroid such that the volume of the ... INTRODUCTION The*hyperplane*conjecture (sometimes called the*slicing*problem) was originally raised*by*J. Bourgain [1] in 1986*and*has attracted*a*lot of attention since. ...##
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Page 4487 of Mathematical Reviews Vol. , Issue 89H
[page]

1989
*
Mathematical Reviews
*

Thus

*a*slightly smaller ball has smaller volume than the unit cube, while all central*hyperplane**slices*of this ball have larger*measure*than the corresponding*slices*of the have*measure*at most cube. ... Soc. 97 (1986), no. 3, 465-473; MR 87g:60018] that all*hyperplane**slices*of*a*unit n-cube 2. ...##
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Some Open Problems in Asymptotic Geometric Analysis

2018
*
Notices of the American Mathematical Society
*

We describe four related open problems in asymptotic geometric analysis: the

doi:10.1090/noti1685
fatcat:7nu3ojp7sjdphcv5wt767dfyum
*hyperplane*conjecture, the isotropic constant conjecture, Sylvester's problem,*and*the simplex conjecture. ...*A*major impulse for the theory is the*hyperplane*conjecture or*slicing*problem. Motivated*by*questions arising in harmonic analysis, it was first formulated*by*J. ... to the*hyperplane*conjecture for isotropic bodies*and*the general case follows*by**a*standard argument. ...##
###
Measures of sections of convex bodies
[article]

2015
*
arXiv
*
pre-print

This article is

arXiv:1511.05525v1
fatcat:tpj6cb77i5d6jmbm355zfkmqu4
*a*survey of recent results on*slicing*inequalities for*convex*bodies. The focus is on the*setting*of arbitrary*measures*in place of volume. ... original*slicing*problem can be extended to the*setting*of arbitrary*measures*. ... Applying stability in Shephard's problem to this pair of bodies, dividing*by*ε*and*sending ε to zero, one gets*a**hyperplane*inequality for surface area (see [K7] ): if L is*a*projection body, then S(L ...##
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Page 419 of Mathematical Reviews Vol. , Issue 2003A
[page]

2003
*
Mathematical Reviews
*

Note that throughout their paper (

*and*this review) saying that two*sets*are equal will mean they agree up to*a**set*of*measure*zero. The*sets*considered are assumed to be compact. ... If for some Hp € , K;*and*K> osculate each other to infinite order along Ho,*and*their cross-sectional areas agree when*sliced**by*planes in /, then K; = K2. ...##
###
Slicing inequalities for subspaces of $L_p$

2015
*
Proceedings of the American Mathematical Society
*

For example, for every k ∈ N there exists

doi:10.1090/proc12708
fatcat:spao4tasijch3gapxihftgwqly
*a*constant C(k) such that for every n ∈ N, k < n, every*convex*k-intersection body (unit ball of*a*normed subspace of L −k ) L in R n*and*every*measure*μ with ... We prove*slicing*inequalities for*measures*of the unit balls of subspaces of L p , −∞ < p < ∞. ... Finally, let us prove*a**slicing*inequality for subspaces of L p , p > 2 with arbitrary*measures*. ...##
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Tight Lower Bounds for Halfspace Range Searching

2012
*
Discrete & Computational Geometry
*

We establish two new lower bounds for the halfspace range searching problem: Given

doi:10.1007/s00454-012-9412-x
fatcat:7npvqbzen5g7ta5guuj4bnao2i
*a**set*of n points in R d , where each point is associated with*a*weight from*a*commutative semigroup, compute the semigroup ... Letting m denote the space requirements, we prove*a*lower bound for general semigroups of Ω n 1−1/(d+1) /m 1/(d+1)*and*for integral semigroups of Ω n/m 1/d . ... Therefore, restricted to the*set*Q, we can interpret this*measure*as*a*probability density*by*dividing it*by*the total*measure*of Q. ...##
###
Tight lower bounds for halfspace range searching

2010
*
Proceedings of the 2010 annual symposium on Computational geometry - SoCG '10
*

We establish two new lower bounds for the halfspace range searching problem: Given

doi:10.1145/1810959.1810964
dblp:conf/compgeom/AryaMX10
fatcat:snrpahrc4bcnjd5mesp6wua4ae
*a**set*of n points in R d , where each point is associated with*a*weight from*a*commutative semigroup, compute the semigroup ... Letting m denote the space requirements, we prove*a*lower bound for general semigroups of Ω n 1−1/(d+1) /m 1/(d+1)*and*for integral semigroups of Ω n/m 1/d . ... Therefore, restricted to the*set*Q, we can interpret this*measure*as*a*probability density*by*dividing it*by*the total*measure*of Q. ...##
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Digital Steiner sets and Matheron semi-groups

2010
*
Image and Vision Computing
*

One finds this law when the γ λ are adjunction openings

doi:10.1016/j.imavis.2009.06.016
fatcat:6yhgr4obu5apdjmjpdpgiq5v4y
*by*Steiner*convex**sets*, i.e.*by*Minkowski sums of segments. ... The conditions under which, in Z n , the law remains valid,*and*the Steiner*sets*are*convex*,*and*connected, are established. ... The supporting*slice*Π(ω, x, y) generated*by*the*hyperplanes*{H(*A*, c) , c ∈ [ c x , c y ]} contains point z [15] . ...##
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Slicing inequalities for subspaces of L_p
[article]

2013
*
arXiv
*
pre-print

We show that the

arXiv:1310.8102v1
fatcat:hxl6po3kmvd2zfrfcxnefc34hi
*hyperplane*conjecture holds for the classes of k-intersection bodies with arbitrary*measures*in place of volume. ... Finally, let us prove*a**slicing*inequality for subspaces of L p , p > 2 with arbitrary*measures*. Theorem 3. ... For any p > 0 there exists*a*constant C(p) such that for any n ∈ N, n > p, any*convex*body L in R n that is the unit ball of*a*normed space embedding in L −p , any 1 ≤ m ≤ n − 1,*and*any*measure*µ with ...##
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Non compact boundaries of complex analytic varieties in Hilbert spaces

2014
*
Complex Manifolds
*

, open subsets of

doi:10.2478/coma-2014-0002
fatcat:wzkch2kfbjhhtcq7my6sfg6vqi
*a*complex Hilbert space H.We deal with the problem*by*cutting with*a*family of complex*hyperplanes**and*applying the already known result for the compact case. ... AbstractWe treat the boundary problem for complex varieties with isolated singularities, of complex dimension greater than or equal to 3, non necessarily compact, which are contained in strongly*convex*... Useful remarks*by*the referee helped us to make clearer proofs*and*correct an erroneous formulation of the rst lemma*and*other misprints. ...##
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Comments on the floating body and the hyperplane conjecture
[article]

2011
*
arXiv
*
pre-print

We provide

arXiv:1102.2570v2
fatcat:jfwsm65ctfgdtkflpe65oileui
*a*reformulation of the*hyperplane*conjecture (the*slicing*problem) in terms of the floating body*and*give upper*and*lower bounds on the logarithmic Hausdorff distance between an arbitrary*convex*... body K⊂R^d*and*the*convex*floating body K_δ inside K. ... The*hyperplane*conjecture (also known as the*slicing*problem) speculates that there exists*a*universal constant c > 0 such that for any d ∈ N*and*any*convex*body K ⊂ R d of unit volume, there exists*a*...##
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A discrete version of Koldobsky's slicing inequality

2017
*
Israel Journal of Mathematics
*

Let #K be

doi:10.1007/s11856-017-1589-5
fatcat:bbhgwhonivhtzhylzk5bj5euby
*a*number of integer lattice points contained in*a**set*K. ... The problem is still open, with the best-to-date estimate of L 1 ≤ O(d 1/4 ) established*by*Klartag [Kl], who improved the previous estimate of Bourgain [Bo2], we refer to [MP]*and*[BGVV] for detailed ... The*slicing*problem of Bourgain [Bo1, Bo2] is, undoubtedly, one of the major open problems in*convex*geometry asking if*a**convex*, origin-symmetric body of volume one must have*a*large (in volume)*hyperplane*...
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