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On the relations between principal eigenvalue and torsional rigidity
[article]

2019
*
arXiv
*
pre-print

*The*full Blaschke-

*Santaló*

*diagram*

*for*λ(Ø)

*and*T(Ø) is obtained in dimension one, while

*for*higher dimensions we provide some bounds. ...

*The*optimisation problem above is considered in

*the*class

*of*all domainsØ, in

*the*class

*of*convex domainsØ,

*and*in

*the*class

*of*thin domains. ...

*The*work

*of*GB

*and*AP is part

*of*

*the*project 2017TEXA3H "Gradient flows, Optimal Transport

*and*Metric Measure Structures" funded by

*the*Italian Ministry

*of*Research

*and*University. ...

##
###
A complete 3-dimensional Blaschke-Santaló diagram
[article]

2014
*
arXiv
*
pre-print

We present a complete 3-dimensional Blaschke-Santal\'o

arXiv:1404.6808v1
fatcat:l37rkhvjwjhjbcmrl56moy6g4e
*diagram**for*planar convex bodies with respect to*the*four classical magnitudes inner*and*outer radius, diameter*and*(minimal) width in euclidean spaces ... Acknowledgements: We would like to thank Viviana Ghiglione*and*Evgeny Zavalnyuk*for*giving crucial hints, as well as Peter Gritzmann, Maria Hernández Cifre,*and*Salvador Segura Gomis*for*always supporting ... Reviving*the*idea*of*Blaschke,*Santaló*proposed in [17]*the*study*of*such*diagrams**for*all triples*of**the*magnitudes*r*,*w*,*D*,*R*, p (perimeter)*and*A (area),*for*a start,*for*planar sets. ...##
###
The missing (A, D, r) diagram
[article]

2021
*
arXiv
*
pre-print

, denoted (A,

arXiv:2005.05749v2
fatcat:jsx755qjrzebfmoozkf5yyyovi
*D*,*r*). ... This allows us to completely determine*the*so-called 2-dimensional Blaschke-*Santaló**diagram**for*planar convex bodies with respect to*the*three magnitudes area, diameter*and*inradius in euclidean spaces ...*The*third author was partially supported by*the*Project "Analysis*and*simulation*of*optimal shapes -application to lifesciences"*of**the*Paris City Hall. ...##
###
The missing (A,D,r) diagram

2022
*
Annales de l'Institut Fourier
*

, denoted (A,

doi:10.5802/aif.3484
fatcat:pcjeynxsq5cpflkipsxbqo4xaa
*D*,*r*). ... This allows us to completely determine*the*so-called 2-dimensional Blaschke-*Santaló**diagram**for*planar convex bodies with respect to*the*three magnitudes area, diameter*and*inradius in euclidean spaces ...*The*third author was partially supported by*the*Project "Analysis*and*simulation*of*optimal shapes -application to lifesciences"*of**the*Paris City Hall. ...##
###
Page 8749 of Mathematical Reviews Vol. , Issue 2000m
[page]

2000
*
Mathematical Reviews
*

(E-MURC; Murcia)

*The**missing**boundaries**of**the*Santal6*diagrams**for**the**cases*(*d*,*w*,*R*)*and*(*w*,*R*,*r*). (English summary) Discrete Comput. Geom. 23 (2000), no. 3, 381-388. ...*For*example,*for*n > 2, A, < c,)*d*(K, L), where*the*constant c; is an explicit function*of*just n*and*max(diam K, diam L). ...##
###
Data depth and floating body
[article]

2018
*
arXiv
*
pre-print

Little known relations

arXiv:1809.10925v1
fatcat:6ejplls2krfmnocsn4dp6zqysu
*of**the*renown concept*of**the*halfspace depth*for*multivariate data with notions from convex*and*affine geometry are discussed. ... Halfspace depth may be regarded as a measure*of*symmetry*for*random vectors. As such,*the*depth stands as a generalization*of*a measure*of*symmetry*for*convex sets, well studied in geometry. ... Stanislav Nagy is supported by*the*grant 18-00522Y*of**the*Czech Science Foundation,*and*by*the*PRIMUS/17/SCI/3 project*of*Charles University. ...##
###
Isoperimetric sets and p-Cheeger sets are in bijection
[article]

2022
*
arXiv
*
pre-print

We study

arXiv:2207.02044v1
fatcat:vtfbuoxh5vbnnaqd4yjxbjbbuq
*the*set-valued map 𝔙:[1/2,+∞)→𝒫((0,|Ω|]) associating to each p*the*set*of*volumes*of*p-Cheeger sets. ... Moreover, when restricted to (1/2, 1) such a map is univalued*and*is in bijection with its image. As a consequence*of*our analysis we derive some fine*boundary*regularity result. ... C :=*w*∈ H 1 (−*r*,*r*) |*w*− f E ∈ H 1 0 (−*r*,*r*),*w*≤ f Ω in (−*r*, Step three: Steps one*and*two above prove*the*validity*of**the*thesis*of*Corollary 2.6*for*sets satisfying either a) or b). ...##
###
Euler Calculus with Applications to Signals and Sensing
[article]

2012
*
arXiv
*
pre-print

This article surveys

arXiv:1202.0275v1
fatcat:puy3hkehzva3vjnqlrdppi7jtm
*the*Euler calculus - an integral calculus based on Euler characteristic -*and*its applications to data, sensing, networks,*and*imaging. ... Yuliy Baryshnikov was*the*first to appreciate applications*of*Euler calculus to sensing*and*is directly or jointly responsible*for*most*of**the*applications appearing in this survey. ... Acknowledgements*The*applications described in this survey would not have been possible without*the*guidance*and*hard work*of*two individuals. ...##
###
Integral geometry on manifolds with boundary and applications
[article]

2018
*
arXiv
*
pre-print

with

arXiv:1806.06088v1
fatcat:rbt5qjck4jhl3d2ifat6bq4uwu
*boundary*. ... We survey recent results on inverse problems*for*geodesic X-ray transforms*and*other linear*and*non-linear geometric inverse problems*for*Riemannian metrics, connections*and*Higgs fields defined on manifolds ... , (12) where*R*is*the*Riemann curvature tensor*of*(M, g), viewed as an operator on*the*bundles N*and*N ⊗ E over SM by*the*actions*R*(x, v)*w*:=*R*x (*w*, v)v,*R*(x, v)(*w*⊗ e) := (*R*x (*w*, v)v) ⊗ e, e ∈ E x , ...##
###
Contact integral geometry and the Heisenberg algebra
[article]

2019
*
arXiv
*
pre-print

Generalizing Weyl's tube formula

arXiv:1712.09313v3
fatcat:bxyk5hbkuvaddi7aazylqiqqum
*and*building on Chern's work, Alesker reinterpreted*the*Lipschitz-Killing curvature integrals as a family*of*valuations (finitely-additive measures with good analytic properties ... Moreover, these valuations generalize to*the*class*of*manifolds equipped with*the*structure*of*a Heisenberg algebra on their cotangent bundle. ... This last piece is*missing*in*the*DH setting. ...##
###
Some Discrete Properties of the Space of Line Transversals to Disjoint Balls
[chapter]

2009
*
IMA Volumes in Mathematics and its Applications
*

We review recent progress in

doi:10.1007/978-1-4419-0999-2_3
fatcat:snteyqtjojh6rjbeui6r4ih65m
*the*special*case**of*disjoint Euclidean balls in*R**d*, more precisely*the*inter-related notions*of*cone*of*directions, geometric permutations*and*Helly-type theorems,*and*discuss ... Attempts to generalize Helly's theorem to sets*of*lines intersecting convex sets led to a series*of*results relating*the*geometry*of*a family*of*sets in*R**d*to*the*structure*of**the*space*of*lines intersecting ...*The*author is grateful to Boris Aronov, Ciprian Borcea, Otfried Cheong, Olivier Devillers, Hazel Everett, Andreas Holmsen*and*Sylvain Petitjean*for*helpful discussions. ...##
###
Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?
[article]

2019
*
arXiv
*
pre-print

While in

arXiv:1911.02120v1
fatcat:rjm5xwq5jjdnjelybl7ilws4my
*case*(i)*the*central limit theorem is valid*for*all*d*≥ 2, it is shown that in*case*(ii)*the*central limit theorem holds*for**d*∈{2,3}*and*fails if*d*≥ 4*and*k=*d*-1 or if*d*≥ 7*and**for*general k. ...*The*central limit problem*for**the*k-dimensional Hausdorff measure*of**the*k-skeleton is approached in two different set-ups: (i)*for*a fixed window*and*growing intensities,*and*(ii)*for*fixed intensity ... Our thanks also go to Tom Kaufmann (Bochum) whose programming skills helped us to verify*the*list*of*permutations we had to use to prove Theorem 5 (a)*and*(b). ...##
###
Beta-star polytopes and hyperbolic stochastic geometry
[article]

2022
*
arXiv
*
pre-print

A beta-star polytope is defined as

arXiv:2109.01035v3
fatcat:kwhrmv4okjckfevqs3i7zvb7ny
*the*convex hull*of*an inhomogeneous Poisson processes on*the*complement*of**the*unit ball in*ℝ*^*d*with density proportional to (||x||^2-1)^β, where ||x|| > 1*and*β>*d*/2. ...*The*general results*for*beta-star polytopes are used to provide explicit formulas*for**the*expected f- vector*of**the*typical hyperbolic Poisson-Voronoi cell*and**the*hyperbolic Poisson zero cell. ... Acknowledgement*The*authors are grateful to Ben Hansen*and*Tobias Müller*for*providing us with*the*simulation*of**the*hyperbolic Poisson-Voronoi tessellation shown in*the*left panel*of*...##
###
Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

2021
*
Probability theory and related fields
*

While in

doi:10.1007/s00440-021-01032-w
fatcat:ntemj6bje5bxrkoe2arjikrpwu
*case*(i)*the*central limit theorem is valid*for*all $$*d*\ge 2$$*d*≥ 2 , it is shown that in*case*(ii)*the*central limit theorem holds*for*$$*d*\in \{2,3\}$$*d*∈ { 2 , 3 }*and*fails if $$*d*\ge 4$$ ...*The*central limit problem*for**the*k-dimensional Hausdorff measure*of**the*k-skeleton is approached in two different set-ups: (i)*for*a fixed window*and*growing intensities,*and*(ii)*for*fixed intensity ... Acknowledgements*The*authors are grateful to an anonymous referee*for*insightful questions, remarks*and*suggestions. This project was initiated when FH was visiting Ruhr University Bochum in March ...##
###
Centro-Affine Differential Geometry and the Log-Minkowski Problem
[article]

2021
*
arXiv
*
pre-print

As a consequence, we resolve

arXiv:2104.12408v3
fatcat:zk4qxahktrccvpmcjphpu53eai
*the*isomorphic version*of**the*log-Minkowski problem:*for*any origin-symmetric convex body K̅ in*ℝ*^n, there exists an origin-symmetric convex body K with K̅⊂ K ⊂ 8 K̅, so that ... In particular, we may use*the*classical argument*of*Lichnerowicz*and*a centro-affine Bochner formula to give a new proof*of**the*Brunn-Minkowski inequality. ... Introduction Here h K*and*S K denote*the*support function*and*surface-area measure*of*K, respectively -we refer to Section 2*for*standard*missing*definitions. ...
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