{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2022,12,29]],"date-time":"2022-12-29T19:11:05Z","timestamp":1672341065825},"reference-count":21,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2009,8,4]],"date-time":"2009-08-04T00:00:00Z","timestamp":1249344000000},"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":[[2009,12]]},"DOI":"10.1007\/s00454-009-9214-y","type":"journal-article","created":{"date-parts":[[2009,8,3]],"date-time":"2009-08-03T19:05:47Z","timestamp":1249326347000},"page":"594-614","source":"Crossref","is-referenced-by-count":4,"title":["Contraction and Expansion of Convex Sets"],"prefix":"10.1007","volume":"42","author":[{"given":"Michael","family":"Langberg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Leonard J.","family":"Schulman","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2009,8,4]]},"reference":[{"issue":"3\/4","key":"9214_CR1","doi-asserted-by":"crossref","first-page":"245","DOI":"10.1007\/BF02574042","volume":"13","author":"N. Alon","year":"1995","unstructured":"Alon, N., Kalai, G.: Bounding the piercing number. Discrete Comput. Geom. 13(3\/4), 245\u2013256 (1995)","journal-title":"Discrete Comput. Geom."},{"issue":"4","key":"9214_CR2","doi-asserted-by":"crossref","first-page":"423","DOI":"10.1007\/BF02711517","volume":"15","author":"N. Amenta","year":"1996","unstructured":"Amenta, N.: A short proof of an interesting Helly-type theorem. Discrete Comput. Geom. 15(4), 423\u2013427 (1996)","journal-title":"Discrete Comput. Geom."},{"issue":"1","key":"9214_CR3","doi-asserted-by":"crossref","first-page":"109","DOI":"10.2307\/2044407","volume":"86","author":"I. B\u00e1r\u00e1ny","year":"1982","unstructured":"B\u00e1r\u00e1ny, I., Pach, M.K.J.: Quantitative Helly-type theorems. Proc. Am. Math. Soc. 86(1), 109\u2013114 (1982)","journal-title":"Proc. Am. Math. Soc."},{"key":"9214_CR4","series-title":"Graduate Studies in Mathematics","doi-asserted-by":"crossref","DOI":"10.1090\/gsm\/054","volume-title":"A Course in Convexity","author":"A. Barvinok","year":"2002","unstructured":"Barvinok, A.: A Course in Convexity. Graduate Studies in Mathematics, vol.\u00a054. Am. Math. Soc., Providence (2002)"},{"key":"9214_CR5","doi-asserted-by":"crossref","unstructured":"Demouth, J., Devillers, O., Glisse, M., Goaoc, X.: Helly-type theorems for approximate covering. In: Proceedings of the Twenty-Fourth Annual Symposium on Computational Geometry, pp. 120\u2013128 (2008)","DOI":"10.1145\/1377676.1377696"},{"issue":"3","key":"9214_CR6","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1016\/0021-9045(74)90120-8","volume":"10","author":"R.M. Dudley","year":"1974","unstructured":"Dudley, R.M.: Metric entropy of some classes of sets with differentiable boundaries. J. Approx. Theory 10(3), 227\u2013236 (1974)","journal-title":"J. Approx. Theory"},{"key":"9214_CR7","doi-asserted-by":"crossref","first-page":"389","DOI":"10.1016\/B978-0-444-89596-7.50017-1","volume-title":"Handbook of Convex Geometry","author":"J. Eckhoff","year":"1993","unstructured":"Eckhoff, J.: Helly, Radon, and Carath\u00e9odory type theorems. In: Gruber, P.M., Wills, J.M. (eds.) Handbook of Convex Geometry, vol. A, pp. 389\u2013448. North-Holland, Amsterdam (1993)"},{"issue":"4","key":"9214_CR8","doi-asserted-by":"crossref","first-page":"215","DOI":"10.1016\/S0925-7721(99)00059-0","volume":"15","author":"A. Efrat","year":"2000","unstructured":"Efrat, A., Katz, M.J., Nielsen, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Comput. Geom. 15(4), 215\u2013227 (2000)","journal-title":"Comput. Geom."},{"key":"9214_CR9","unstructured":"Gao, J., Langberg, M., Schulman, L.J.: Clustering lines: classification of incomplete data. Manuscript (2006)"},{"key":"9214_CR10","doi-asserted-by":"crossref","first-page":"319","DOI":"10.1016\/B978-0-444-89596-7.50015-8","volume-title":"Handbook of Convex Geometry","author":"P.M. Gruber","year":"1993","unstructured":"Gruber, P.M.: Aspects of approximation of convex bodies. In: Handbook of Convex Geometry, vol.\u00a0A, pp. 319\u2013345. North-Holland, Amsterdam (1993)"},{"issue":"4","key":"9214_CR11","doi-asserted-by":"crossref","first-page":"607","DOI":"10.2307\/2034254","volume":"12","author":"B. Grunbaum","year":"1961","unstructured":"Grunbaum, B., Motzkin, T.S.: On components in some families of sets. Proc. Am. Math. Soc. 12(4), 607\u2013613 (1961)","journal-title":"Proc. Am. Math. Soc."},{"key":"9214_CR12","first-page":"175","volume":"32","author":"E. Helly","year":"1923","unstructured":"Helly, E.: \u00dcber Mengen konvexer K\u00f6rper mit gemeinschaftlichen Punkten. Jahresber. Dtsch. Math.-Ver. 32, 175\u2013176 (1923)","journal-title":"Jahresber. Dtsch. Math.-Ver."},{"key":"9214_CR13","first-page":"187","volume-title":"Studies and Essays Presented to R. Courant on his 60th Birthday","author":"F. John","year":"1948","unstructured":"John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday, pp. 187\u2013204. Interscience, New York (1948)"},{"issue":"3","key":"9214_CR14","doi-asserted-by":"crossref","first-page":"551","DOI":"10.1112\/jtopol\/jtn010","volume":"1","author":"G. Kalai","year":"2008","unstructured":"Kalai, G., Meshulam, R.: Leray numbers of projections and a topological Helly-type theorem. J.\u00a0Topol. 1(3), 551\u2013556 (2008)","journal-title":"J.\u00a0Topol."},{"key":"9214_CR15","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1112\/S0025579300002370","volume":"15","author":"D.G. Larman","year":"1968","unstructured":"Larman, D.G.: Helly type properties of unions of convex sets. Mathematika 15, 53\u201359 (1968)","journal-title":"Mathematika"},{"issue":"1","key":"9214_CR16","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1007\/PL00009305","volume":"18","author":"J. Matou\u0161ek","year":"1997","unstructured":"Matou\u0161ek, J.: A Helly-type theorem for unions of convex sets. Comput. Geom. 18(1), 1\u201312 (1997)","journal-title":"Comput. Geom."},{"key":"9214_CR17","unstructured":"Morris, H.: Two pigeonhole principles and unions of convexly disjoint sets. Ph.D. Thesis, Calif. Inst. of Techn., Calif. (1973)"},{"key":"9214_CR18","doi-asserted-by":"crossref","first-page":"245","DOI":"10.4153\/CJM-1950-022-8","volume":"2","author":"H. Rademacher","year":"1950","unstructured":"Rademacher, H., Schoenberg, I.J.: Helly\u2019s theorem on convex domains and Tchebycheff\u2019s approximation problem. Can. J. Math. 2, 245\u2013256 (1950)","journal-title":"Can. J. Math."},{"key":"9214_CR19","first-page":"53","volume":"2","author":"L.G. Shnirelman","year":"1938","unstructured":"Shnirelman, L.G.: On uniform approximations. Izv. Akad. Nauk SSSR Ser. Mat. 2, 53\u201360 (1938)","journal-title":"Izv. Akad. Nauk SSSR Ser. Mat."},{"key":"9214_CR20","doi-asserted-by":"crossref","first-page":"353","DOI":"10.1016\/0925-7721(93)90007-S","volume":"3","author":"A.F. Stappen van der","year":"1993","unstructured":"van der Stappen, A.F., Halperin, D., Overmars, M.H.: The complexity of the free space for a robot moving amidst fat obstacles. Comput. Geom. Theory Appl. 3, 353\u2013373 (1993)","journal-title":"Comput. Geom. Theory Appl."},{"key":"9214_CR21","first-page":"63","volume-title":"Handbook of Discrete and Computational Geometry","author":"R. Wenger","year":"1997","unstructured":"Wenger, R.: Helly-type theorems and geometric transversals. In: Handbook of Discrete and Computational Geometry, pp. 63\u201382. CRC Press, Boca Raton (1997)"}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-009-9214-y.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s00454-009-9214-y\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s00454-009-9214-y","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2019,5,28]],"date-time":"2019-05-28T23:47:38Z","timestamp":1559087258000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s00454-009-9214-y"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2009,8,4]]},"references-count":21,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2009,12]]}},"alternative-id":["9214"],"URL":"https:\/\/doi.org\/10.1007\/s00454-009-9214-y","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[2009,8,4]]}}}