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Slicing convex sets and measures by a hyperplane [article]

Imre Barany, Alfredo Hubard, Jesus Jeronimo
2010 arXiv   pre-print
,α_d ∈ [0, 1], we give 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.  ... 
arXiv:1010.6279v1 fatcat:zkf4m3epmbh4jequgjdmcotroq

Slicing Convex Sets and Measures by a Hyperplane

Imre Bárány, Alfredo Hubard, Jesús Jerónimo
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.  ... 
doi:10.1007/s00454-007-9021-2 fatcat:s2hknbmycvctfh4b3ucbbiupui

Entropy and the hyperplane conjecture in convex geometry

Sergey Bobkov, Mokshay Madiman
2010 2010 IEEE International Symposium on Information Theory  
The 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.  ... 
doi:10.1109/isit.2010.5513619 dblp:conf/isit/BobkovM10 fatcat:k6uvkmqxwraytkastxf4yy2h4u

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.  ... 

Some Open Problems in Asymptotic Geometric Analysis

Bo'Az Klartag, Elisabeth Werner
2018 Notices of the American Mathematical Society  
We describe four related open problems in asymptotic geometric analysis: the 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.  ... 
doi:10.1090/noti1685 fatcat:7nu3ojp7sjdphcv5wt767dfyum

Measures of sections of convex bodies [article]

Alexander Koldobsky
2015 arXiv   pre-print
This article is 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  ... 
arXiv:1511.05525v1 fatcat:tpj6cb77i5d6jmbm355zfkmqu4

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 $L_p$

Alexander Koldobsky
2015 Proceedings of the American Mathematical Society  
For example, for every k ∈ N there exists 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.  ... 
doi:10.1090/proc12708 fatcat:spao4tasijch3gapxihftgwqly

Tight Lower Bounds for Halfspace Range Searching

Sunil Arya, David M. Mount, Jian Xia
2012 Discrete & Computational Geometry  
We establish two new lower bounds for the halfspace range searching problem: Given 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.  ... 
doi:10.1007/s00454-012-9412-x fatcat:7npvqbzen5g7ta5guuj4bnao2i

Tight lower bounds for halfspace range searching

Sunil Arya, David M. Mount, Jian Xia
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 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.  ... 
doi:10.1145/1810959.1810964 dblp:conf/compgeom/AryaMX10 fatcat:snrpahrc4bcnjd5mesp6wua4ae

Digital Steiner sets and Matheron semi-groups

Jean Serra
2010 Image and Vision Computing  
One finds this law when the γ λ are adjunction openings 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] .  ... 
doi:10.1016/j.imavis.2009.06.016 fatcat:6yhgr4obu5apdjmjpdpgiq5v4y

Slicing inequalities for subspaces of L_p [article]

Alexander Koldobsky
2013 arXiv   pre-print
We show that the 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  ... 
arXiv:1310.8102v1 fatcat:hxl6po3kmvd2zfrfcxnefc34hi

Non compact boundaries of complex analytic varieties in Hilbert spaces

Samuele Mongodi, Alberto Saracco
2014 Complex Manifolds  
, open subsets of 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.  ... 
doi:10.2478/coma-2014-0002 fatcat:wzkch2kfbjhhtcq7my6sfg6vqi

Comments on the floating body and the hyperplane conjecture [article]

Daniel Fresen
2011 arXiv   pre-print
We provide 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  ... 
arXiv:1102.2570v2 fatcat:jfwsm65ctfgdtkflpe65oileui

A discrete version of Koldobsky's slicing inequality

Matthew Alexander, Martin Henk, Artem Zvavitch
2017 Israel Journal of Mathematics  
Let #K be 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  ... 
doi:10.1007/s11856-017-1589-5 fatcat:bbhgwhonivhtzhylzk5bj5euby
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