{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T08:14:56Z","timestamp":1772266496962,"version":"3.50.1"},"reference-count":48,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T00:00:00Z","timestamp":1600214400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T00:00:00Z","timestamp":1600214400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100002790","name":"Canadian Network for Research and Innovation in Machining Technology, Natural Sciences and Engineering Research Council of Canada","doi-asserted-by":"publisher","award":["2015-0674"],"award-info":[{"award-number":["2015-0674"]}],"id":[{"id":"10.13039\/501100002790","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001807","name":"FAPESP","doi-asserted-by":"crossref","award":["2013\/03447-6"],"award-info":[{"award-number":["2013\/03447-6"]}],"id":[{"id":"10.13039\/501100001807","id-type":"DOI","asserted-by":"crossref"}]},{"name":"DFG","award":["SFB\/TRR 191"],"award-info":[{"award-number":["SFB\/TRR 191"]}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Math. Program."],"published-print":{"date-parts":[[2022,2]]},"abstract":"<jats:title>Abstract<\/jats:title><jats:p>We introduce the cone of completely positive functions, a subset of the cone of positive-type functions, and use it to fully characterize maximum-density distance-avoiding sets as the optimal solutions of a convex optimization problem. As a consequence of this characterization, it is possible to reprove and improve many results concerning distance-avoiding sets on the sphere and in Euclidean space.<\/jats:p>","DOI":"10.1007\/s10107-020-01562-6","type":"journal-article","created":{"date-parts":[[2020,9,16]],"date-time":"2020-09-16T22:50:37Z","timestamp":1600296637000},"page":"487-558","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":5,"title":["Complete positivity and distance-avoiding sets"],"prefix":"10.1007","volume":"191","author":[{"given":"Evan","family":"DeCorte","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Fernando M\u00e1rio de Oliveira","family":"Filho","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Frank","family":"Vallentin","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2020,9,16]]},"reference":[{"key":"1562_CR1","doi-asserted-by":"crossref","first-page":"645","DOI":"10.1007\/s00039-009-0013-7","volume":"19","author":"C Bachoc","year":"2009","unstructured":"Bachoc, C., Nebe, G., de Oliveira Filho, F.M., Vallentin, F.: Lower bounds for measurable chromatic numbers. Geom. Funct. Anal. 19, 645\u2013661 (2009)","journal-title":"Geom. Funct. Anal."},{"key":"1562_CR2","doi-asserted-by":"crossref","first-page":"783","DOI":"10.1007\/s00454-015-9668-z","volume":"53","author":"C Bachoc","year":"2015","unstructured":"Bachoc, C., Passuello, A., Thiery, A.: The density of sets avoiding distance 1 in Euclidean space. Discrete Comput. Geom. 53, 783\u2013808 (2015)","journal-title":"Discrete Comput. Geom."},{"key":"1562_CR3","volume-title":"A Course in Convexity. Graduate Studies in Mathematics 54","author":"A Barvinok","year":"2002","unstructured":"Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics 54. American Mathematical Society, Providence, RI (2002)"},{"key":"1562_CR4","doi-asserted-by":"crossref","first-page":"647","DOI":"10.2307\/1969252","volume":"42","author":"S Bochner","year":"1941","unstructured":"Bochner, S.: Hilbert distances and positive definite functions. Ann. Math. 42, 647\u2013656 (1941)","journal-title":"Ann. Math."},{"key":"1562_CR5","doi-asserted-by":"crossref","first-page":"307","DOI":"10.1007\/BF02764959","volume":"54","author":"J Bourgain","year":"1986","unstructured":"Bourgain, J.: A Szemer\u00e9di type theorem for sets of positive density in $${\\mathbb{R}}^k$$. Isr. J. Math. 54, 307\u2013316 (1986)","journal-title":"Isr. J. Math."},{"key":"1562_CR6","doi-asserted-by":"crossref","first-page":"668","DOI":"10.1007\/s00039-008-0673-8","volume":"18","author":"B Bukh","year":"2008","unstructured":"Bukh, B.: Measurable sets with excluded distances. Geom. Funct. Anal. 18, 668\u2013697 (2008)","journal-title":"Geom. Funct. Anal."},{"key":"1562_CR7","doi-asserted-by":"crossref","first-page":"689","DOI":"10.4007\/annals.2003.157.689","volume":"157","author":"H Cohn","year":"2003","unstructured":"Cohn, H., Elkies, N.: New upper bounds on sphere packings I. Ann. Math. 157, 689\u2013714 (2003)","journal-title":"Ann. Math."},{"key":"1562_CR8","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.4007\/annals.2017.185.3.8","volume":"185","author":"H Cohn","year":"2017","unstructured":"Cohn, H., Kumar, A., Miller, S.D., Radchenko, D., Viazovska, M.: The sphere packing problem in dimension 24. Ann. Math. 185, 1017\u20131033 (2017)","journal-title":"Ann. Math."},{"key":"1562_CR9","doi-asserted-by":"crossref","first-page":"75","DOI":"10.1007\/s10898-006-9037-9","volume":"37","author":"E de Klerk","year":"2007","unstructured":"de Klerk, E., Pasechnik, D.V.: A linear programming reformulation of the standard quadratic optimization problem. J. Global Optim. 37, 75\u201384 (2007)","journal-title":"J. Global Optim."},{"key":"1562_CR10","doi-asserted-by":"crossref","first-page":"1944","DOI":"10.1137\/15M103114X","volume":"26","author":"E de Klerk","year":"2016","unstructured":"de Klerk, E., Vallentin, F.: On the Turing model complexity of interior point methods for semidefinite programming. SIAM J. Optim. 26, 1944\u20131961 (2016)","journal-title":"SIAM J. Optim."},{"key":"1562_CR11","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1007\/s10107-014-0843-4","volume":"151","author":"D de Laat","year":"2015","unstructured":"de Laat, D., Vallentin, F.: A semidefinite programming hierarchy for packing problems in discrete geometry. Math. Program. Ser. B 151, 529\u2013553 (2015)","journal-title":"Math. Program. Ser. B"},{"key":"1562_CR12","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.4171\/JEMS\/236","volume":"12","author":"FM de Oliveira Filho","year":"2010","unstructured":"de Oliveira Filho, F.M., Vallentin, F.: Fourier analysis, linear programming, and densities of distance-avoiding sets in $$\\mathbb{R }^n$$. J. Eur. Math. Soc. 12, 1417\u20131428 (2010)","journal-title":"J. Eur. Math. Soc."},{"key":"1562_CR13","doi-asserted-by":"crossref","first-page":"295","DOI":"10.1007\/s10711-012-9814-1","volume":"167","author":"FM de Oliveira Filho","year":"2013","unstructured":"de Oliveira Filho, F.M., Vallentin, F.: A quantitative version of Steinhaus\u2019s theorem for compact, connected, rank-one symmetric spaces. Geom. Dedicata 167, 295\u2013307 (2013)","journal-title":"Geom. Dedicata"},{"key":"1562_CR14","doi-asserted-by":"crossref","first-page":"785","DOI":"10.1112\/S0025579319000160","volume":"65","author":"FM de Oliveira Filho","year":"2019","unstructured":"de Oliveira Filho, F.M., Vallentin, F.: A counterexample to a conjecture of Larman and Rogers on sets avoiding distance 1. Mathematika 65, 785 (2019)","journal-title":"Mathematika"},{"key":"1562_CR15","doi-asserted-by":"crossref","first-page":"6095","DOI":"10.1093\/imrn\/rnv319","volume":"20","author":"E DeCorte","year":"2016","unstructured":"DeCorte, E., Pikhurko, O.: Spherical sets avoiding a prescribed set of angles. Int. Math. Res. Not. 20, 6095\u20136117 (2016)","journal-title":"Int. Math. Res. Not."},{"key":"1562_CR16","doi-asserted-by":"crossref","first-page":"363","DOI":"10.1007\/BF03187604","volume":"6","author":"P Delsarte","year":"1977","unstructured":"Delsarte, P., Goethals, J.M., Seidel, J.J.: Spherical codes and designs. Geom. Dedicata 6, 363\u2013388 (1977)","journal-title":"Geom. Dedicata"},{"key":"1562_CR17","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-04295-9","volume-title":"Geometry of Cuts and Metrics. Algorithms and Combinatorics 15","author":"MM Deza","year":"1997","unstructured":"Deza, M.M., Laurent, M.: Geometry of Cuts and Metrics. Algorithms and Combinatorics 15. Springer, Berlin (1997)"},{"key":"1562_CR18","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1007\/s10107-015-0974-2","volume":"160","author":"C Dobre","year":"2016","unstructured":"Dobre, C., D\u00fcr, M.E., Frerick, L., Vallentin, F.: A copositive formulation of the stability number of infinite graphs. Math. Program. Ser. A 160, 65\u201383 (2016)","journal-title":"Math. Program. Ser. A"},{"key":"1562_CR19","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1016\/0097-3165(81)90014-5","volume":"31","author":"KJ Falconer","year":"1981","unstructured":"Falconer, K.J.: The realization of distances in measurable subsets covering $${\\mathbb{R}}^n$$. J. Comb. Theory Ser. A 31, 184\u2013189 (1981)","journal-title":"J. Comb. Theory Ser. A"},{"key":"1562_CR20","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1112\/blms\/18.5.471","volume":"18","author":"KJ Falconer","year":"1986","unstructured":"Falconer, K.J., Marstrand, J.M.: Plane sets with positive density at infinity contain all large distances. Bull. Lond. Math. Soc. 18, 471\u2013474 (1986)","journal-title":"Bull. Lond. Math. Soc."},{"key":"1562_CR21","volume-title":"Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften","author":"H Federer","year":"1969","unstructured":"Federer, H.: Geometric Measure Theory. Die Grundlehren der mathematischen Wissenschaften, vol. 153. Springer, New York (1969)"},{"key":"1562_CR22","volume-title":"Real Analysis: Modern Techniques and Their Applications","author":"GB Folland","year":"1999","unstructured":"Folland, G.B.: Real Analysis: Modern Techniques and Their Applications. Wiley, New York (1999)"},{"key":"1562_CR23","doi-asserted-by":"crossref","first-page":"184","DOI":"10.1007\/978-3-642-72905-8_13","volume-title":"Mathematics of Ramsey Theory","author":"H Furstenberg","year":"1990","unstructured":"Furstenberg, H., Katznelson, Y., Weiss, B.: Ergodic theory and configurations in sets of positive density. In: Ne\u0161et\u0159il, J., R\u00f6dl, V. (eds.) Mathematics of Ramsey Theory, pp. 184\u2013198. Springer, Berlin (1990)"},{"key":"1562_CR24","doi-asserted-by":"crossref","first-page":"411","DOI":"10.2140\/pjm.1958.8.411","volume":"8","author":"JW Gaddum","year":"1958","unstructured":"Gaddum, J.W.: Linear inequalities and quadratic forms. Pac. J. Math. 8, 411\u2013414 (1958)","journal-title":"Pac. J. Math."},{"key":"1562_CR25","doi-asserted-by":"crossref","DOI":"10.1007\/978-3-642-97881-4","volume-title":"Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics 2","author":"M Gr\u00f6tschel","year":"1988","unstructured":"Gr\u00f6tschel, M., Lov\u00e1\u00e1sz, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization, Algorithms and Combinatorics 2. Springer, Berlin (1988)"},{"key":"1562_CR26","doi-asserted-by":"crossref","unstructured":"Kalai, G.: Some old and new problems in combinatorial geometry\u00a0I: around Borsuk\u2019s problem. In: Surveys in combinatorics\u00a02015, London Mathematical Society Lecture Note Series\u00a0424. Cambridge University Press, Cambridge, pp.\u00a0147\u2013174 (2015)","DOI":"10.1017\/CBO9781316106853.005"},{"key":"1562_CR27","first-page":"85","volume-title":"Complexity of Computer Computations (Proceedings of a symposium on the Complexity of Computer Computations, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, 1972)","author":"RM Karp","year":"1972","unstructured":"Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R.E., Thatcher, J.W. (eds.) Complexity of Computer Computations (Proceedings of a symposium on the Complexity of Computer Computations, IBM Thomas J. Watson Research Center, Yorktown Heights, New York, 1972), pp. 85\u2013103. Plenum Press, New York (1972)"},{"key":"1562_CR28","doi-asserted-by":"crossref","first-page":"642","DOI":"10.1007\/s00454-015-9751-5","volume":"55","author":"T Keleti","year":"2016","unstructured":"Keleti, T., Matolcsi, M., de Oliveira Filho, F.M., Ruzsa, I.Z.: Better bounds for planar sets avoiding unit distances. Discrete Comput. Geom. 55, 642\u2013661 (2016)","journal-title":"Discrete Comput. Geom."},{"key":"1562_CR29","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1112\/S0025579300004903","volume":"19","author":"DG Larman","year":"1972","unstructured":"Larman, D.G., Rogers, C.A.: The realization of distances within sets in Euclidean space. Mathematika 19, 1\u201324 (1972)","journal-title":"Mathematika"},{"key":"1562_CR30","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1109\/TIT.1979.1055985","volume":"IT\u201325","author":"L Lov\u00e1sz","year":"1979","unstructured":"Lov\u00e1sz, L.: On the Shannon capacity of a graph. IEEE Trans. Inf. Theory IT\u201325, 1\u20137 (1979)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1562_CR31","doi-asserted-by":"crossref","first-page":"933","DOI":"10.1016\/j.jctb.2006.05.002","volume":"96","author":"L Lov\u00e1sz","year":"2006","unstructured":"Lov\u00e1sz, L., Szegedy, B.: Limits of dense graph sequences. J. Comb. Theory Ser. B 96, 933\u2013957 (2006)","journal-title":"J. Comb. Theory Ser. B"},{"key":"1562_CR32","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511623813","volume-title":"Geometry of Sets and Measures in Euclidean Space: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics 44","author":"P Mattila","year":"1995","unstructured":"Mattila, P.: Geometry of Sets and Measures in Euclidean Space: Fractals and Rectifiability, Cambridge Studies in Advanced Mathematics 44. Cambridge University Press, Cambridge (1995)"},{"key":"1562_CR33","first-page":"134","volume":"3","author":"RJ McEliece","year":"1978","unstructured":"McEliece, R.J., Rodemich, E.R., Rumsey, H.C.: The Lov\u00e1sz bound and some generalizations. J. Comb. Inf. Syst. Sci. 3, 134\u2013152 (1978)","journal-title":"J. Comb. Inf. Syst. Sci."},{"key":"1562_CR34","doi-asserted-by":"crossref","first-page":"293","DOI":"10.1016\/S0001-8708(76)80002-3","volume":"21","author":"J Milnor","year":"1976","unstructured":"Milnor, J.: Curvatures of left invariant metrics on Lee groups. Adv. Math. 21, 293\u2013329 (1976)","journal-title":"Adv. Math."},{"key":"1562_CR35","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/0166-218X(91)90071-4","volume":"31","author":"WOJ Moser","year":"1991","unstructured":"Moser, W.O.J.: Problems, problems, problems. Discrete Appl. Math. 31, 201\u2013225 (1991)","journal-title":"Discrete Appl. Math."},{"key":"1562_CR36","doi-asserted-by":"crossref","first-page":"533","DOI":"10.4153\/CJM-1965-053-6","volume":"17","author":"TS Motzkin","year":"1965","unstructured":"Motzkin, T.S., Straus, E.G.: Maxima for graphs and a new proof of a theorem of Tur\u00e1n. Can. J. Math. 17, 533\u2013540 (1965)","journal-title":"Can. J. Math."},{"key":"1562_CR37","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1007\/BF01589101","volume":"45","author":"M Padberg","year":"1989","unstructured":"Padberg, M.: The Boolean quadric polytope: some characteristics, facets and relatives. Math. Program. Ser. B 45, 139\u2013172 (1989)","journal-title":"Math. Program. Ser. B"},{"key":"1562_CR38","volume-title":"Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness","author":"M Reed","year":"1975","unstructured":"Reed, M., Simon, B.: Methods of Modern Mathematical Physics II: Fourier Analysis, Self-adjointness. Academic Press, New York (1975)"},{"key":"1562_CR39","doi-asserted-by":"crossref","first-page":"811","DOI":"10.2307\/1968466","volume":"39","author":"IJ Schoenberg","year":"1938","unstructured":"Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811\u2013841 (1938)","journal-title":"Ann. Math."},{"key":"1562_CR40","doi-asserted-by":"crossref","first-page":"96","DOI":"10.1215\/S0012-7094-42-00908-6","volume":"9","author":"IJ Schoenberg","year":"1942","unstructured":"Schoenberg, I.J.: Positive definite functions on spheres. Duke Math. J. 9, 96\u2013108 (1942)","journal-title":"Duke Math. J."},{"key":"1562_CR41","doi-asserted-by":"crossref","first-page":"425","DOI":"10.1109\/TIT.1979.1056072","volume":"IT\u201325","author":"A Schrijver","year":"1979","unstructured":"Schrijver, A.: A comparison of the Delsarte and Lov\u00e1sz bounds. IEEE Trans. Inf. Theory IT\u201325, 425\u2013429 (1979)","journal-title":"IEEE Trans. Inf. Theory"},{"key":"1562_CR42","volume-title":"Combinatorial Optimization: Polyhedra and Efficiency","author":"A Schrijver","year":"2003","unstructured":"Schrijver, A.: Combinatorial Optimization: Polyhedra and Efficiency, vol. B. Springer, Berlin (2003)"},{"key":"1562_CR43","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511910135","volume-title":"Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics 187","author":"B Simon","year":"2011","unstructured":"Simon, B.: Convexity: An Analytic Viewpoint, Cambridge Tracts in Mathematics 187. Cambridge University Press, Cambridge (2011)"},{"key":"1562_CR44","volume-title":"Orthogonal Polynomials. American Mathematical Society Colloquium Publications Volume XXIII","author":"G Szeg\u00f6","year":"1975","unstructured":"Szeg\u00f6, G.: Orthogonal Polynomials. American Mathematical Society Colloquium Publications Volume XXIII, 4th edn. American Mathematical Society, Providence (1975)","edition":"4"},{"key":"1562_CR45","first-page":"646","volume-title":"Paul Erd\u0151s and His Mathematics II Bolyai Society Mathematical Studies\u00a011, J\u00e1nos Bolyai Mathematical Society, Budapest","author":"LA Sz\u00e9kely","year":"2002","unstructured":"Sz\u00e9kely, L.A.: Erd\u0151s on unit distances and the Szemer\u00e9di\u2013Trotter theorems. In: Hal\u00e1sz, G., Lov\u00e1sz, L., Simonovits, M., S\u00f3s, V.T. (eds.) Paul Erd\u0151s and His Mathematics II Bolyai Society Mathematical Studies\u00a011, J\u00e1nos Bolyai Mathematical Society, Budapest, pp. 646\u2013666. Springer, Berlin (2002)"},{"key":"1562_CR46","doi-asserted-by":"crossref","first-page":"991","DOI":"10.4007\/annals.2017.185.3.7","volume":"185","author":"MS Viazovska","year":"2017","unstructured":"Viazovska, M.S.: The sphere packing problem in dimension 8. Ann. Math. 185, 991\u20131015 (2017)","journal-title":"Ann. Math."},{"key":"1562_CR47","volume-title":"A Treatise on the Theory of Bessel Functions","author":"GN Watson","year":"1922","unstructured":"Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)"},{"key":"1562_CR48","doi-asserted-by":"crossref","first-page":"1101","DOI":"10.1080\/00029890.1974.11993744","volume":"10","author":"HS Witsenhausen","year":"1974","unstructured":"Witsenhausen, H.S.: Spherical sets without orthogonal point pairs. Am. Math. Mon. 10, 1101\u20131102 (1974)","journal-title":"Am. Math. Mon."}],"container-title":["Mathematical Programming"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01562-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10107-020-01562-6\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10107-020-01562-6.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2022,2,22]],"date-time":"2022-02-22T16:05:13Z","timestamp":1645545913000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10107-020-01562-6"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2020,9,16]]},"references-count":48,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2022,2]]}},"alternative-id":["1562"],"URL":"https:\/\/doi.org\/10.1007\/s10107-020-01562-6","relation":{},"ISSN":["0025-5610","1436-4646"],"issn-type":[{"value":"0025-5610","type":"print"},{"value":"1436-4646","type":"electronic"}],"subject":[],"published":{"date-parts":[[2020,9,16]]},"assertion":[{"value":"11 March 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"1 September 2020","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 September 2020","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}