Fetching the paper…
Reading the bibliography…
We bound the variance and other moments of a random vector based on the range of its realizations, thus generalizing inequalities of Popoviciu (1935) and Bhatia and Davis (2000) concerning measures on the line to several dimensions.
Über die kleinste Kugel, die eine räumliche Figur einschliesst
Heinrich Jung · 1901
Earlier work this paper cites.
Isodiametry, variance, and regular simplices from particle interactions
Tongseok Lim and Robert J. McCann · 1907
Earlier work this paper cites.
Sur les équations algébriques ayant toutes leurs racines réelles
Tiberiu Popoviciu · 1935
Earlier work this paper cites.
Helly’s theorem and its relatives
Ludwig Danzer, Branko Grünbaum and Victor Klee · 1963
Earlier work this paper cites.
Geometric Measure Theory
Herbert Federer · 1969
Earlier work this paper cites.
Convex Analysis
R. Tyrrell Rockafellar · 1970
Earlier work this paper cites.
New bounds on the number of unit spheres that can touch a unit sphere in n n dimensions
A. M. Odlyzko and N. J. A. Sloane · 1979
Earlier work this paper cites.
Rearrangements and convexity of level sets in PDE
Bernhard Kawohl · 1985
Earlier work this paper cites.
Mathematical theory of statistics. Statistical experiments and asymptotic decision theory
Helmut Strasser · 1985
Earlier work this paper cites.
Convex bodies: the Brunn-Minkowski theory
Rolf Schneider · 1993
Earlier work this paper cites.
A convexity principle for interacting gases
Robert J. McCann · 1997
Cited alongside, same era.
Some comments and a bibliography on the Laguerre-Samuelson inequality with extensions and applications in statistics and matrix theory
Shane T. Jensen and George P. H. Styan · 1999
Cited alongside, same era.
A better bound on the variance
Rajendra Bhatia and Chandler Davis · 2000
Cited alongside, same era.
The sphere packing problem in dimension 8
Maryna S. Viazovska · 2001
Cited alongside, same era.
New upper bounds on sphere packings. I
Henry Cohn and Noam Elkies · 2003
Cited alongside, same era.
Techniques of variational analysis
Jonathan M. Borwein and Qiji J. Zhu · 2005
Cited alongside, same era.
Stability analysis of flock and mill rings for second order models in swarming
G. Albi, D. Balagué, J. A. Carrillo, and J. von Brecht · 2014
Later among the works it cites.
The derivation of swarming models: mean-field limit and Wasserstein distances
José Antonio Carrillo, Young-Pil Choi, and Maxime Hauray · 2014
Later among the works it cites.
Ring patterns and their bifurcations in a nonlocal model of biological swarms
Andrea L. Bertozzi, Theodore Kolokolnikov, Hui Sun, David Uminsky, and James von Brecht · 2015
Later among the works it cites.
On minimizers of interaction functionals with competing attractive and repulsive potentials
Rustum Choksi, Razvan C. Fetecau, and Ihsan Topaloglu · 2015
Later among the works it cites.
Geometry of minimizers for the interaction energy with mildly repulsive potentials
J. A. Carrillo, A. Figalli, and F. S. Patacchini · 2017
Later among the works it cites.
alphaXiv searches the wider corpus for related work and actual follow-ups.
alphaXiv is searching for related work…
Stable stationary states of non-local interaction equations
Klemens Fellner and Gaël Raoul · 2010
Cited alongside, same era.
Dimensionality of local minimizers of the interaction energy
D. Balagué, J. A. Carrillo, T. Laurent, and G. Raoul · 2013
Cited alongside, same era.
Variational methods in the presence of symmetry
Jonathan M. Borwein and Qiji J. Zhu · 2013
Cited alongside, same era.
Five lectures on optimal transportation: geometry, regularity, and applications. In Analysis and Geometry of Metric Measure Spaces: Lecture Notes of the Séminaire de Mathématiques Supérieure (SMS) Montréal 2011
Robert J. McCann and Nestor Guillen · 2013
Cited alongside, same era.
Nonlinear aggregation-diffusion equations: radial symmetry and long time asymptotics
J. A. Carrillo, S. Hittmeir, B. Volzone, and Y. Yao
Cited in the paper.
Explicit equilibrium solutions for the aggregation equation with power-law potentials
José A. Carrillo and Yanghong Huang · 2017
Later among the works it cites.
Extended generalised variances, with applications
Luc Pronzato, Henry P. Wynn and Anatoly A. Zhigljavsky, Anatoly A · 2017
Later among the works it cites.
Nonlocal shape optimization via interactions of attractive and repulsive potentials
Almut Burchard, Rustum Choksi, and Ihsan Topaloglu · 2018
Later among the works it cites.
A “liquid-solid” phase transition in a simple model for swarming, based on the “no flat-spots” theorem for subharmonic functions
Rupert L. Frank and Elliott H. Lieb · 2018
Later among the works it cites.
Uniqueness and radial symmetry of minimizers for a nonlocal variational problem
Orlando Lopes · 2019
Later among the works it cites.