{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,25]],"date-time":"2026-06-25T20:21:00Z","timestamp":1782418860723,"version":"3.54.5"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2010,4,2]],"date-time":"2010-04-02T00:00:00Z","timestamp":1270166400000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Discrete Comput Geom"],"published-print":{"date-parts":[[2010,9]]},"DOI":"10.1007\/s00454-010-9256-1","type":"journal-article","created":{"date-parts":[[2010,4,1]],"date-time":"2010-04-01T15:23:38Z","timestamp":1270135418000},"page":"281-307","source":"Crossref","is-referenced-by-count":65,"title":["Bregman Voronoi Diagrams"],"prefix":"10.1007","volume":"44","author":[{"given":"Jean-Daniel","family":"Boissonnat","sequence":"first","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Frank","family":"Nielsen","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Richard","family":"Nock","sequence":"additional","affiliation":[],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"297","published-online":{"date-parts":[[2010,4,2]]},"reference":[{"key":"9256_CR1","isbn-type":"print","volume-title":"Methods of Information Geometry","author":"S. Amari","year":"2000","unstructured":"Amari, S., Nagaoka, H.: Methods of Information Geometry. Oxford University Press, London (2000). ISBN 13 9780821843024","ISBN":"https:\/\/id.crossref.org\/isbn\/9780821843024"},{"key":"9256_CR2","first-page":"345","volume":"43","author":"C. Atkinson","year":"1981","unstructured":"Atkinson, C., Mitchell, F.S.: Rao\u2019s distance measure. Sankhya Indian J. Stat., Ser. A 43, 345\u2013365 (1981)","journal-title":"Sankhya Indian J. Stat., Ser. A"},{"issue":"1","key":"9256_CR3","doi-asserted-by":"crossref","first-page":"78","DOI":"10.1137\/0216006","volume":"16","author":"F. Aurenhammer","year":"1987","unstructured":"Aurenhammer, F.: Power diagrams: Properties, algorithms and applications. SIAM J. Comput. 16(1), 78\u201396 (1987)","journal-title":"SIAM J. Comput."},{"key":"9256_CR4","first-page":"53","volume-title":"4th Annual Symposium on Theoretical Aspects of Computer Sciences (STACS)","author":"F. Aurenhammer","year":"1987","unstructured":"Aurenhammer, F., Imai, H.: Geometric relations among Voronoi diagrams. In: 4th Annual Symposium on Theoretical Aspects of Computer Sciences (STACS), pp. 53\u201365. Springer, London (1987)"},{"key":"9256_CR5","doi-asserted-by":"crossref","first-page":"201","DOI":"10.1016\/B978-044482537-7\/50006-1","volume-title":"Handbook of Computational Geometry","author":"F. Aurenhammer","year":"2000","unstructured":"Aurenhammer, F., Klein, R.: Voronoi diagrams. In: Sack, J., Urrutia, G. (eds.) Handbook of Computational Geometry, pp. 201\u2013290. Elsevier, Amsterdam (2000). Chap.\u00a0V"},{"key":"9256_CR6","first-page":"1705","volume":"6","author":"A. Banerjee","year":"2005","unstructured":"Banerjee, A., Merugu, S., Dhillon, I.S., Ghosh, J.: Clustering with Bregman divergences. J.\u00a0Mach. Learn. Res. 6, 1705\u20131749 (2005)","journal-title":"J.\u00a0Mach. Learn. Res."},{"key":"9256_CR7","first-page":"125","volume":"2","author":"A. Ben-Hur","year":"2002","unstructured":"Ben-Hur, A., Horn, D., Siegelmann, H.T., Vapnik, V.: Support vector clustering. J.\u00a0Mach. Learn. Res. 2, 125\u2013137 (2002)","journal-title":"J.\u00a0Mach. Learn. Res."},{"key":"9256_CR8","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9781139172998","volume-title":"Algorithmic Geometry","author":"J.-D. Boissonnat","year":"1998","unstructured":"Boissonnat, J.-D., Yvinec, M.: Algorithmic Geometry. Cambridge University Press, New York (1998)"},{"key":"9256_CR9","series-title":"Mathematics and Visualization","first-page":"67","volume-title":"Effective Computational Geometry for Curves and Surfaces","author":"J.-D. Boissonnat","year":"2007","unstructured":"Boissonnat, J.-D., Wormser, C., Yvinec, M.: Curved Voronoi diagrams. In: Boissonnat, J.-D., Teillaud, M. (eds.) Effective Computational Geometry for Curves and Surfaces. Mathematics and Visualization, pp. 67\u2013116. Springer, Berlin (2007)"},{"issue":"1","key":"9256_CR10","doi-asserted-by":"crossref","first-page":"37","DOI":"10.1007\/s00454-009-9175-1","volume":"42","author":"J.-D. Boissonnat","year":"2009","unstructured":"Boissonnat, J.-D., Guibas, L.J., Oudot, S.Y.: Manifold reconstruction in arbitrary dimensions using witness complexes. Discrete Comput. Geom. 42(1), 37\u201370 (2009)","journal-title":"Discrete Comput. Geom."},{"key":"9256_CR11","doi-asserted-by":"crossref","first-page":"200","DOI":"10.1016\/0041-5553(67)90040-7","volume":"7","author":"L.M. Bregman","year":"1967","unstructured":"Bregman, L.M.: The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comput. Math. Math. Phys. 7, 200\u2013217 (1967)","journal-title":"USSR Comput. Math. Math. Phys."},{"issue":"4","key":"9256_CR12","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1007\/BF02570718","volume":"14","author":"H. Br\u00f6nnimann","year":"1995","unstructured":"Br\u00f6nnimann, H., Goodrich, M.T.: Optimal set covers in finite VC-dimension. Discrete Comput. Geom. 14(4), 463\u2013479 (1995)","journal-title":"Discrete Comput. Geom."},{"key":"9256_CR13","volume-title":"Parallel Optimization: Theory, Algorithms and Applications","author":"Y.A. Censor","year":"1997","unstructured":"Censor, Y.A., Zenios, S.A.: Parallel Optimization: Theory, Algorithms and Applications. Oxford University Press, London (1997)"},{"key":"9256_CR14","first-page":"282","volume-title":"SODA \u201995: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms","author":"T.M. Chan","year":"1995","unstructured":"Chan, T.M., Snoeyink, J., Yap, C.-K.: Output-sensitive construction of polytopes in four dimensions and clipped Voronoi diagrams in three. In: SODA \u201995: Proceedings of the Sixth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 282\u2013291. SIAM, Philadelphia (1995)"},{"key":"9256_CR15","doi-asserted-by":"crossref","first-page":"377","DOI":"10.1007\/BF02573985","volume":"10","author":"B. Chazelle","year":"1993","unstructured":"Chazelle, B.: An optimal convex hull algorithm in any fixed dimension. Discrete Comput. Geom. 10, 377\u2013409 (1993)","journal-title":"Discrete Comput. Geom."},{"key":"9256_CR16","doi-asserted-by":"crossref","DOI":"10.1017\/CBO9780511626371","volume-title":"The Discrepancy Method","author":"B. Chazelle","year":"2000","unstructured":"Chazelle, B.: The Discrepancy Method. Cambridge University Press, Cambridge (2000)"},{"key":"9256_CR17","doi-asserted-by":"crossref","first-page":"387","DOI":"10.1007\/BF02187740","volume":"4","author":"K. Clarkson","year":"1989","unstructured":"Clarkson, K., Shor, P.: Applications of random sampling in computational geometry, ii. Discrete Comput. Geom. 4, 387\u2013421 (1989)","journal-title":"Discrete Comput. Geom."},{"key":"9256_CR18","doi-asserted-by":"crossref","first-page":"2032","DOI":"10.1214\/aos\/1176348385","volume":"19","author":"I. Csisz\u00e1r","year":"1991","unstructured":"Csisz\u00e1r, I.: Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems. Ann. Stat. 19, 2032\u20132066 (1991)","journal-title":"Ann. Stat."},{"issue":"1","key":"9256_CR19","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1007\/s10711-008-9261-1","volume":"135","author":"V. Silva de","year":"2008","unstructured":"de Silva, V.: A\u00a0weak characterisation of the Delaunay triangulation. Geom. Dedic. 135(1), 39\u201364 (2008)","journal-title":"Geom. Dedic."},{"key":"9256_CR20","volume-title":"Principia Philosophiae","author":"R. Descartes","year":"1644","unstructured":"Descartes, R.: Principia Philosophiae. Ludovicus Elzevirius, Amsterdam (1644)"},{"issue":"2","key":"9256_CR21","doi-asserted-by":"crossref","first-page":"358","DOI":"10.1016\/j.ipl.2005.03.010","volume":"95","author":"G. Even","year":"2005","unstructured":"Even, G., Rawitz, D., Shahar, S.: Hitting sets when the VC-dimension is small. Inf. Process. Lett. 95(2), 358\u2013362 (2005)","journal-title":"Inf. Process. Lett."},{"key":"9256_CR22","doi-asserted-by":"crossref","unstructured":"Inaba, M., Imai, H.: Geometric clustering models for multimedia databases. In: Proceedings of the 10th Canadian Conference on Computational Geometry (CCCG\u201998) (1998)","DOI":"10.1007\/3-540-49292-5_51"},{"key":"9256_CR23","unstructured":"Inaba, M., Imai, H.: Geometric clustering for multiplicative mixtures of distributions in exponential families. In: Proceedings of the 12th Canadian Conference on Computational Geometry (CCCG\u201900) (2000)"},{"key":"9256_CR24","series-title":"Lecture Notes in Computer Science","isbn-type":"print","doi-asserted-by":"crossref","DOI":"10.1007\/3-540-52055-4","volume-title":"Concrete and Abstract Voronoi Diagrams","author":"R. Klein","year":"1989","unstructured":"Klein, R.: Concrete and Abstract Voronoi Diagrams. Lecture Notes in Computer Science, vol.\u00a0400. Springer, Berlin (1989). ISBN 3-540-52055-4","ISBN":"https:\/\/id.crossref.org\/isbn\/3540520554"},{"key":"9256_CR25","first-page":"191","volume-title":"Proc. 19th Symposium on Computational Geometry (SoCG)","author":"F. Labelle","year":"2003","unstructured":"Labelle, F., Shewchuk, J.R.: Anisotropic Voronoi diagrams and guaranteed-quality anisotropic mesh generation. In: Proc. 19th Symposium on Computational Geometry (SoCG), pp. 191\u2013200. ACM, New York (2003)"},{"key":"9256_CR26","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4613-0039-7","volume-title":"Lectures on Discrete Geometry","author":"J. Matousek","year":"2002","unstructured":"Matousek, J.: Lectures on Discrete Geometry. Springer, Berlin (2002)"},{"key":"9256_CR27","doi-asserted-by":"crossref","first-page":"179","DOI":"10.1016\/0095-8956(71)90042-6","volume":"10","author":"P. McMullen","year":"1971","unstructured":"McMullen, P.: The maximum numbers of faces of a convex polytope. J.\u00a0Comb. Theory, Ser. B 10, 179\u2013184 (1971)","journal-title":"J.\u00a0Comb. Theory, Ser. B"},{"key":"9256_CR28","unstructured":"Nielsen, F.: Visual Computing: Geometry, Graphics, and Vision. Charles River Media\/Thomson Delmar Learning (2005)"},{"key":"9256_CR29","unstructured":"Nielsen, F., Garcia, V.: Statistical exponential families: A\u00a0digest with flash cards. arXiv:0911.4863 (2009)"},{"key":"9256_CR30","volume-title":"International Symposium on Voronoi Diagrams (ISVD)","author":"F. Nielsen","year":"2009","unstructured":"Nielsen, F., Nock, R.: The dual Voronoi diagrams with respect to representational Bregman divergences. In: International Symposium on Voronoi Diagrams (ISVD), DTU Lyngby, Denmark, June 2009. IEEE Press, New York (2009)"},{"key":"9256_CR31","doi-asserted-by":"crossref","unstructured":"Nielsen, F., Nock, R.: Hyperbolic Voronoi diagrams made easy. arXiv:0903.3287 (March 2009)","DOI":"10.1109\/ICCSA.2010.37"},{"key":"9256_CR32","doi-asserted-by":"crossref","first-page":"154","DOI":"10.1007\/978-3-540-87481-2_11","volume-title":"ECML PKDD \u201908: Proceedings of the European Conference on Machine Learning and Knowledge Discovery in Databases\u2014Part II","author":"R. Nock","year":"2008","unstructured":"Nock, R., Luosto, P., Kivinen, J.: Mixed Bregman clustering with approximation guarantees. In: ECML PKDD \u201908: Proceedings of the European Conference on Machine Learning and Knowledge Discovery in Databases\u2014Part II, pp. 154\u2013169. Springer, Berlin (2008)"},{"key":"9256_CR33","first-page":"463","volume-title":"Proc. 13th Symposium on Computational Geometry (SoCG)","author":"K. Onishi","year":"1997","unstructured":"Onishi, K., Imai, H.: Voronoi diagram in statistical parametric space by Kullback\u2013Leibler divergence. In: Proc. 13th Symposium on Computational Geometry (SoCG), pp. 463\u2013465. ACM, New York (1997)"},{"key":"9256_CR34","unstructured":"Onishi, K., Imai, H.: Voronoi diagrams for an exponential family of probability distributions in information geometry. In: Japan\u2013Korea Joint Workshop on Algorithms and Computation, pp.\u00a01\u20138 (1997)"},{"key":"9256_CR35","doi-asserted-by":"crossref","first-page":"189","DOI":"10.1007\/BF02574375","volume":"12","author":"V.T. Rajan","year":"1994","unstructured":"Rajan, V.T.: Optimality of the Delaunay triangulation in \u211d d . Discrete Comput. Geom. 12, 189\u2013202 (1994)","journal-title":"Discrete Comput. Geom."},{"key":"9256_CR36","doi-asserted-by":"crossref","DOI":"10.1515\/9781400873173","volume-title":"Convex Analysis","author":"R.T. Rockafellar","year":"1970","unstructured":"Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)"},{"key":"9256_CR37","first-page":"403","volume-title":"Proc. 14th Symposium on Computational Geometry (SoCG)","author":"K. Sadakane","year":"1998","unstructured":"Sadakane, K., Imai, H., Onishi, K., Inaba, M., Takeuchi, F., Imai, K.: Voronoi diagrams by divergences with additive weights. In: Proc. 14th Symposium on Computational Geometry (SoCG), pp. 403\u2013404. ACM Press, New York (1998)"},{"key":"9256_CR38","unstructured":"Teillaud, M., Devillers, O., Meiser, S.: The space of spheres, a\u00a0geometric tool to unify duality results on Voronoi diagrams. Technical Report Rapports de Recherche No. 1620, Institut National de Recherche en Informatique et en Automatique (1992)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-010-9256-1.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-010-9256-1\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-010-9256-1","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,31]],"date-time":"2023-05-31T17:56:49Z","timestamp":1685555809000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-010-9256-1"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2010,4,2]]},"references-count":38,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2010,9]]}},"alternative-id":["9256"],"URL":"https:\/\/doi.org\/10.1007\/s00454-010-9256-1","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2010,4,2]]}}}