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

Michiel van den Berg, Giuseppe Buttazzo, Aldo Pratelli
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.  ... 
arXiv:1910.14593v2 fatcat:4sburxltzzafditqjjpr472taa

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

René Brandenberg, Bernardo González Merino
2014 arXiv   pre-print
We present a complete 3-dimensional Blaschke-Santal\'o 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.  ... 
arXiv:1404.6808v1 fatcat:l37rkhvjwjhjbcmrl56moy6g4e

The missing (A, D, r) diagram [article]

Alexandre Delyon
2021 arXiv   pre-print
, denoted (A, 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.  ... 
arXiv:2005.05749v2 fatcat:jsx755qjrzebfmoozkf5yyyovi

The missing (A,D,r) diagram

Alexandre Delyon, Antoine Henrot, Yannick Privat
2022 Annales de l'Institut Fourier  
, denoted (A, 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.  ... 
doi:10.5802/aif.3484 fatcat:pcjeynxsq5cpflkipsxbqo4xaa

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]

Stanislav Nagy, Carsten Schuett, Elisabeth M. Werner
2018 arXiv   pre-print
Little known relations 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.  ... 
arXiv:1809.10925v1 fatcat:6ejplls2krfmnocsn4dp6zqysu

Isoperimetric sets and p-Cheeger sets are in bijection [article]

Marco Caroccia, Giorgio Saracco
2022 arXiv   pre-print
We study 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).  ... 
arXiv:2207.02044v1 fatcat:vtfbuoxh5vbnnaqd4yjxbjbbuq

Euler Calculus with Applications to Signals and Sensing [article]

Justin Curry, Robert Ghrist, Michael Robinson
2012 arXiv   pre-print
This article surveys 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.  ... 
arXiv:1202.0275v1 fatcat:puy3hkehzva3vjnqlrdppi7jtm

Integral geometry on manifolds with boundary and applications [article]

Joonas Ilmavirta, François Monard
2018 arXiv   pre-print
with 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 ,  ... 
arXiv:1806.06088v1 fatcat:rbt5qjck4jhl3d2ifat6bq4uwu

Contact integral geometry and the Heisenberg algebra [article]

Dmitry Faifman
2019 arXiv   pre-print
Generalizing Weyl's tube formula 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.  ... 
arXiv:1712.09313v3 fatcat:bxyk5hbkuvaddi7aazylqiqqum

Some Discrete Properties of the Space of Line Transversals to Disjoint Balls [chapter]

Xavier Goaoc
2009 IMA Volumes in Mathematics and its Applications  
We review recent progress in 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.  ... 
doi:10.1007/978-1-4419-0999-2_3 fatcat:snteyqtjojh6rjbeui6r4ih65m

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space? [article]

Felix Herold and Daniel Hug and Christoph Thähle
2019 arXiv   pre-print
While in 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).  ... 
arXiv:1911.02120v1 fatcat:rjm5xwq5jjdnjelybl7ilws4my

Beta-star polytopes and hyperbolic stochastic geometry [article]

Thomas Godland and Zakhar Kabluchko and Christoph Thäle
2022 arXiv   pre-print
A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in ^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  ... 
arXiv:2109.01035v3 fatcat:kwhrmv4okjckfevqs3i7zvb7ny

Does a central limit theorem hold for the k-skeleton of Poisson hyperplanes in hyperbolic space?

Felix Herold, Daniel Hug, Christoph Thäle
2021 Probability theory and related fields  
While in 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  ... 
doi:10.1007/s00440-021-01032-w fatcat:ntemj6bje5bxrkoe2arjikrpwu

Centro-Affine Differential Geometry and the Log-Minkowski Problem [article]

Emanuel Milman
2021 arXiv   pre-print
As a consequence, we resolve the isomorphic version of the log-Minkowski problem: for any origin-symmetric convex body K̅ in ^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.  ... 
arXiv:2104.12408v3 fatcat:zk4qxahktrccvpmcjphpu53eai
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