{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,1]],"date-time":"2026-04-01T10:44:16Z","timestamp":1775040256455,"version":"3.50.1"},"reference-count":45,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2022,10,4]],"date-time":"2022-10-04T00:00:00Z","timestamp":1664841600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2022,10,4]],"date-time":"2022-10-04T00:00:00Z","timestamp":1664841600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"name":"National Science Foundation","award":["DMS-1439786"],"award-info":[{"award-number":["DMS-1439786"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2023,3]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We consider measures of covering and separation that are expressed through maxima and minima of distances between points of an hypersphere. We investigate the behavior of these measures when applied to a sample of independent and uniformly distributed points. In particular, we derive their asymptotic distributions when the number of points diverges. These results can be useful as a benchmark against which deterministic point sets can be evaluated. Whenever possible, we supplement the rigorous derivation of these limiting distributions with some heuristic reasonings based on extreme value theory. As a by-product, we provide a proof for a conjecture on the hole radius associated to a facet of the convex hull of points distributed on the hypersphere.<\/jats:p>","DOI":"10.1007\/s00454-022-00413-9","type":"journal-article","created":{"date-parts":[[2022,10,4]],"date-time":"2022-10-04T19:02:17Z","timestamp":1664910137000},"page":"285-299","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Asymptotic Distributions of Covering and Separation Measures on the Hypersphere"],"prefix":"10.1007","volume":"69","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1646-3547","authenticated-orcid":false,"given":"Raffaello","family":"Seri","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2022,10,4]]},"reference":[{"key":"413_CR1","doi-asserted-by":"crossref","unstructured":"Alishahi, K., Zamani, M.: The spherical ensemble and uniform distribution of points on the sphere. Electron. J. Probab. 20, # 23 (2015)","DOI":"10.1214\/EJP.v20-3733"},{"issue":"2","key":"413_CR2","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1007\/s00607-001-1434-9","volume":"68","author":"B Bajnok","year":"2002","unstructured":"Bajnok, B., Damelin, S.B., Li, J., Mullen, G.L.: A constructive finite field method for scattering points on the surface of $$d$$-dimensional spheres. Computing 68(2), 97\u2013109 (2002)","journal-title":"Computing"},{"issue":"2","key":"413_CR3","doi-asserted-by":"publisher","first-page":"349","DOI":"10.2140\/pjm.1973.46.349","volume":"46","author":"D Barnette","year":"1973","unstructured":"Barnette, D.: A proof of the lower bound conjecture for convex polytopes. Pac. J. Math. 46(2), 349\u2013354 (1973)","journal-title":"Pac. J. Math."},{"issue":"1","key":"413_CR4","doi-asserted-by":"publisher","first-page":"163","DOI":"10.1007\/s00365-018-9426-6","volume":"48","author":"C Beltr\u00e1n","year":"2018","unstructured":"Beltr\u00e1n, C., Etayo, U.: The projective ensemble and distribution of points in odd-dimensional spheres. Constr. Approx. 48(1), 163\u2013182 (2018)","journal-title":"Constr. Approx."},{"key":"413_CR5","doi-asserted-by":"crossref","unstructured":"Beltr\u00e1n, C., Etayo, U.: A generalization of the spherical ensemble to even-dimensional spheres. J. Math. Anal. Appl. 475(2), 1073\u20131092 (2019)","DOI":"10.1016\/j.jmaa.2019.03.004"},{"key":"413_CR6","doi-asserted-by":"publisher","first-page":"76","DOI":"10.1016\/j.jco.2016.08.001","volume":"37","author":"C Beltr\u00e1n","year":"2016","unstructured":"Beltr\u00e1n, C., Marzo, J., Ortega-Cerd\u00e0, J.: Energy and discrepancy of rotationally invariant determinantal point processes in high dimensional spheres. J. Complex. 37, 76\u2013109 (2016)","journal-title":"J. Complex."},{"key":"413_CR7","doi-asserted-by":"crossref","unstructured":"Bobrowski, O., Schulte, M., Yogeshwaran, D.: Poisson process approximation under stabilization and Palm coupling (2021). arXiv:2104.13261","DOI":"10.5802\/ahl.156"},{"key":"413_CR8","doi-asserted-by":"crossref","unstructured":"Bourgain, J., Sarnak, P., Rudnick, Z.: Local statistics of lattice points on the sphere. In: Modern Trends in Constructive Function Theory (Nashville 2014). Contemporary Mathematics, vol. 661, pp. 269\u2013282. American Mathematical Society, Providence (2016)","DOI":"10.1090\/conm\/661\/13287"},{"key":"413_CR9","doi-asserted-by":"crossref","unstructured":"Brandolini, L., Choirat, C., Colzani, L., Gigante, G., Seri, R., Travaglini, G.: Quadrature rules and distribution of points on manifolds. Ann. Sc. Norm. Super. Pisa Cl. Sci. 13(4), 889\u2013923 (2014)","DOI":"10.2422\/2036-2145.201103_007"},{"issue":"2","key":"413_CR10","doi-asserted-by":"publisher","first-page":"782","DOI":"10.1016\/j.jmaa.2015.05.079","volume":"431","author":"JS Brauchart","year":"2015","unstructured":"Brauchart, J.S., Dick, J., Saff, E.B., Sloan, I.H., Wang, Y.G., Womersley, R.S.: Covering of spheres by spherical caps and worst-case error for equal weight cubature in Sobolev spaces. J. Math. Anal. Appl. 431(2), 782\u2013811 (2015)","journal-title":"J. Math. Anal. Appl."},{"issue":"3","key":"413_CR11","doi-asserted-by":"publisher","first-page":"647","DOI":"10.1007\/s11118-014-9387-8","volume":"41","author":"JS Brauchart","year":"2014","unstructured":"Brauchart, J.S., Dragnev, P.D., Saff, E.B.: Riesz external field problems on the hypersphere and optimal point separation. Potential Anal. 41(3), 647\u2013678 (2014)","journal-title":"Potential Anal."},{"issue":"1","key":"413_CR12","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s00365-006-0629-4","volume":"25","author":"JS Brauchart","year":"2007","unstructured":"Brauchart, J.S., Hesse, K.: Numerical integration over spheres of arbitrary dimension. Constr. Approx. 25(1), 41\u201371 (2007)","journal-title":"Constr. Approx."},{"issue":"1","key":"413_CR13","doi-asserted-by":"publisher","first-page":"62","DOI":"10.1080\/10586458.2016.1226209","volume":"27","author":"JS Brauchart","year":"2018","unstructured":"Brauchart, J.S., Reznikov, A.B., Saff, E.B., Sloan, I.H., Wang, Y.G., Womersley, R.S.: Random point sets on the sphere\u2014hole radii, covering, and separation. Exp. Math. 27(1), 62\u201381 (2018)","journal-title":"Exp. Math."},{"issue":"2","key":"413_CR14","doi-asserted-by":"publisher","first-page":"225","DOI":"10.1007\/BF01455988","volume":"271","author":"C Buchta","year":"1985","unstructured":"Buchta, C., M\u00fcller, J., Tichy, R.F.: Stochastical approximation of convex bodies. Math. Ann. 271(2), 225\u2013235 (1985)","journal-title":"Math. Ann."},{"key":"413_CR15","first-page":"1837","volume":"14","author":"T Cai","year":"2013","unstructured":"Cai, T., Fan, J., Jiang, T.: Distributions of angles in random packing on spheres. J. Mach. Learn. Res. 14, 1837\u20131864 (2013)","journal-title":"J. Mach. Learn. Res."},{"issue":"3","key":"413_CR16","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1007\/s10687-014-0184-y","volume":"17","author":"P Calka","year":"2014","unstructured":"Calka, P., Chenavier, N.: Extreme values for characteristic radii of a Poisson\u2013Voronoi tessellation. Extremes 17(3), 359\u2013385 (2014)","journal-title":"Extremes"},{"issue":"3","key":"413_CR17","doi-asserted-by":"publisher","first-page":"880","DOI":"10.1017\/jpr.2021.92","volume":"59","author":"N Chenavier","year":"2022","unstructured":"Chenavier, N., Henze, N., Otto, M.: Limit laws for large $$k$$th-nearest neighbor balls. J. Appl. Probab. 59(3), 880\u2013894 (2022)","journal-title":"J. Appl. Probab."},{"issue":"284","key":"413_CR18","doi-asserted-by":"publisher","first-page":"2137","DOI":"10.1090\/S0025-5718-2013-02698-1","volume":"82","author":"C Choirat","year":"2013","unstructured":"Choirat, C., Seri, R.: Computational aspects of Cui\u2013Freeden statistics for equidistribution on the sphere. Math. Comp. 82(284), 2137\u20132156 (2013)","journal-title":"Math. Comp."},{"issue":"2","key":"413_CR19","doi-asserted-by":"publisher","first-page":"216","DOI":"10.1016\/j.jco.2012.11.005","volume":"29","author":"C Choirat","year":"2013","unstructured":"Choirat, C., Seri, R.: Numerical properties of generalized discrepancies on spheres of arbitrary dimension. J. Complex. 29(2), 216\u2013235 (2013)","journal-title":"J. Complex."},{"issue":"2","key":"413_CR20","doi-asserted-by":"publisher","first-page":"595","DOI":"10.1137\/S1064827595281344","volume":"18","author":"J Cui","year":"1997","unstructured":"Cui, J., Freeden, W.: Equidistribution on the sphere. SIAM J. Sci. Comput. 18(2), 595\u2013609 (1997)","journal-title":"SIAM J. Sci. Comput."},{"issue":"3","key":"413_CR21","doi-asserted-by":"publisher","first-page":"231","DOI":"10.1016\/S0885-064X(02)00006-7","volume":"19","author":"SB Damelin","year":"2003","unstructured":"Damelin, S.B., Grabner, P.J.: Energy functionals, numerical integration and asymptotic equidistribution on the sphere. J. Complex. 19(3), 231\u2013246 (2003)","journal-title":"J. Complex."},{"key":"413_CR22","doi-asserted-by":"crossref","unstructured":"Damelin, S.B., Grabner, P.J.: Corrigendum to \u201cEnergy functionals, numerical integration and asymptotic equidistribution on the sphere\u201d: [J. Complexity 19 (2003) 231\u2013246]. J. Complex. 20(6), 883\u2013884 (2004)","DOI":"10.1016\/j.jco.2004.06.003"},{"issue":"6","key":"413_CR23","doi-asserted-by":"publisher","first-page":"845","DOI":"10.1016\/j.jco.2005.04.005","volume":"21","author":"SB Damelin","year":"2005","unstructured":"Damelin, S.B., Maymeskul, V.: On point energies, separation radius and mesh norm for $$s$$-extremal configurations on compact sets in $${\\mathbb{R}}^n$$. J. Complex. 21(6), 845\u2013863 (2005)","journal-title":"J. Complex."},{"issue":"4","key":"413_CR24","doi-asserted-by":"publisher","first-page":"745","DOI":"10.1214\/aop\/1176994663","volume":"8","author":"P Diaconis","year":"1980","unstructured":"Diaconis, P., Freedman, D.: Finite exchangeable sequences. Ann. Probab. 8(4), 745\u2013764 (1980)","journal-title":"Ann. Probab."},{"key":"413_CR25","doi-asserted-by":"crossref","unstructured":"Embrechts, P., Kl\u00fcppelberg, C., Mikosch, T.: Modelling Extremal Events. Applications of Mathematics (New York), vol. 33. Springer, Berlin (1997)","DOI":"10.1007\/978-3-642-33483-2"},{"key":"413_CR26","volume-title":"Statistical Analysis of Spherical Data","author":"NI Fisher","year":"1993","unstructured":"Fisher, N.I., Lewis, T., Embleton, B.J.J.: Statistical Analysis of Spherical Data. Cambridge University Press, Cambridge (1993)"},{"key":"413_CR27","doi-asserted-by":"crossref","unstructured":"Gin\u00e9 E.: Invariant tests for uniformity on compact Riemannian manifolds based on Sobolev norms. Ann. Stat. 3(6), 1243\u20131266 (1975)","DOI":"10.1214\/aos\/1176343283"},{"issue":"3","key":"413_CR28","first-page":"407","volume":"8","author":"A Gut","year":"2002","unstructured":"Gut, A.: On the moment problem. Bernoulli 8(3), 407\u2013421 (2002)","journal-title":"Bernoulli"},{"issue":"2","key":"413_CR29","doi-asserted-by":"publisher","first-page":"344","DOI":"10.2307\/3213486","volume":"19","author":"N Henze","year":"1982","unstructured":"Henze, N.: The limit distribution for maxima of \u201cweighted\u2019\u2019 $$r$$th-nearest-neighbour distances. J. Appl. Probab. 19(2), 344\u2013354 (1982)","journal-title":"J. Appl. Probab."},{"issue":"4","key":"413_CR30","doi-asserted-by":"publisher","first-page":"245","DOI":"10.1007\/BF02056931","volume":"30","author":"N Henze","year":"1983","unstructured":"Henze, N.: Ein asymptotischer Satz \u00fcber den maximalen Minimalabstand von unabh\u00e4ngigen Zufallsvektoren mit Anwendung auf einen Anpassungstest im R$$^{p}$$ und auf der Kugel. Metrika 30(4), 245\u2013259 (1983)","journal-title":"Metrika"},{"issue":"2","key":"413_CR31","doi-asserted-by":"publisher","first-page":"659","DOI":"10.1017\/jpr.2018.41","volume":"55","author":"N Henze","year":"2018","unstructured":"Henze, N.: On the consistency of the spacings test for multivariate uniformity, including on manifolds. J. Appl. Probab. 55(2), 659\u2013665 (2018)","journal-title":"J. Appl. Probab."},{"issue":"1","key":"413_CR32","doi-asserted-by":"publisher","first-page":"274","DOI":"10.1214\/aop\/1176992269","volume":"15","author":"S Janson","year":"1987","unstructured":"Janson, S.: Maximal spacings in several dimensions. Ann. Probab. 15(1), 274\u2013280 (1987)","journal-title":"Ann. Probab."},{"issue":"2","key":"413_CR33","doi-asserted-by":"publisher","first-page":"523","DOI":"10.1090\/S0002-9947-98-02119-9","volume":"350","author":"ABJ Kuijlaars","year":"1998","unstructured":"Kuijlaars, A.B.J., Saff, E.B.: Asymptotics for minimal discrete energy on the sphere. Trans. Am. Math. Soc. 350(2), 523\u2013538 (1998)","journal-title":"Trans. Am. Math. Soc."},{"key":"413_CR34","volume-title":"NIST Handbook of Mathematical Functions","year":"2010","unstructured":"Olver, F.W.J., Lozier, D.W., Boisvert, R.F., Clark, C.W. (eds.): NIST Handbook of Mathematical Functions. Cambridge University Press, Cambridge (2010)"},{"issue":"1","key":"413_CR35","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1214\/aos\/1176344075","volume":"6","author":"MJ Prentice","year":"1978","unstructured":"Prentice, M.J.: On invariant tests of uniformity for directions and orientations. Ann. Stat. 6(1), 169\u2013176 (1978)","journal-title":"Ann. Stat."},{"key":"413_CR36","doi-asserted-by":"crossref","unstructured":"Prentice, M.J.: Correction to: \u201cOn invariant tests of uniformity for directions and orientations\u201d. Ann. Stat. 7(4), 926 (1979)","DOI":"10.1214\/aos\/1176344744"},{"issue":"6","key":"413_CR37","doi-asserted-by":"publisher","first-page":"647","DOI":"10.4310\/MRL.1994.v1.n6.a3","volume":"1","author":"EA Rakhmanov","year":"1994","unstructured":"Rakhmanov, E.A., Saff, E.B., Zhou, Y.M.: Minimal discrete energy on the sphere. Math. Res. Lett. 1(6), 647\u2013662 (1994)","journal-title":"Math. Res. Lett."},{"key":"413_CR38","doi-asserted-by":"crossref","unstructured":"Rakhmanov, E.A., Saff, E.B., Zhou, Y.M.: Electrons on the sphere. In: 2nd International Conference on Computational Methods and Function Theory (Penang 1994). Ser. Approx. Decompos., vol.\u00a05, pp. 293\u2013309. World Scientific, River Edge (1995)","DOI":"10.1142\/9789814533232_0022"},{"issue":"19","key":"413_CR39","doi-asserted-by":"publisher","first-page":"6065","DOI":"10.1093\/imrn\/rnv342","volume":"2016","author":"A Reznikov","year":"2016","unstructured":"Reznikov, A., Saff, E.B.: The covering radius of randomly distributed points on a manifold. Int. Math. Res. Not. 2016(19), 6065\u20136094 (2016)","journal-title":"Int. Math. Res. Not."},{"issue":"1\u20133","key":"413_CR40","doi-asserted-by":"publisher","first-page":"469","DOI":"10.1007\/s00454-007-9012-3","volume":"39","author":"RM Richardson","year":"2008","unstructured":"Richardson, R.M., Vu, V.H., Wu, L.: An inscribing model for random polytopes. Discrete Comput. Geom. 39(1\u20133), 469\u2013499 (2008)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"413_CR41","doi-asserted-by":"publisher","first-page":"5","DOI":"10.1007\/BF03024331","volume":"19","author":"EB Saff","year":"1997","unstructured":"Saff, E.B., Kuijlaars, A.B.J.: Distributing many points on a sphere. Math. Intell. 19(1), 5\u201311 (1997)","journal-title":"Math. Intell."},{"key":"413_CR42","doi-asserted-by":"crossref","unstructured":"Schulte, M., Th\u00e4le, C.: Poisson point process convergence and extreme values in stochastic geometry. In: Stochastic Analysis for Poisson Point Processes. Bocconi Springer Ser., vol.\u00a07, pp. 255\u2013294. Bocconi University Press (2016)","DOI":"10.1007\/978-3-319-05233-5_8"},{"key":"413_CR43","doi-asserted-by":"crossref","unstructured":"Shorack, G.R.: Probability for Statisticians. Springer Texts in Statistics. Springer, Cham (2017)","DOI":"10.1007\/978-3-319-52207-4"},{"key":"413_CR44","unstructured":"Stemeseder, J.: Random Polytopes with Vertices on the Boundary of a Smooth Convex Body. PhD thesis, University of Salzburg (2014)"},{"key":"413_CR45","series-title":"Cambridge Series in Statistical and Probabilistic Mathematics","doi-asserted-by":"publisher","DOI":"10.1017\/CBO9780511802256","volume-title":"Asymptotic Statistics","author":"AW van der Vaart","year":"1998","unstructured":"van der Vaart, A.W.: Asymptotic Statistics. Cambridge Series in Statistical and Probabilistic Mathematics, vol. 3. Cambridge University Press, Cambridge (1998)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-022-00413-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00454-022-00413-9\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-022-00413-9.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,2,6]],"date-time":"2023-02-06T15:05:50Z","timestamp":1675695950000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00454-022-00413-9"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,10,4]]},"references-count":45,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2023,3]]}},"alternative-id":["413"],"URL":"https:\/\/doi.org\/10.1007\/s00454-022-00413-9","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2022,10,4]]},"assertion":[{"value":"4 November 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 February 2022","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"14 March 2022","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 October 2022","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The author declares that he has no conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}]}}