{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,11]],"date-time":"2026-03-11T17:38:17Z","timestamp":1773250697600,"version":"3.50.1"},"reference-count":19,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[1988,12,1]],"date-time":"1988-12-01T00:00:00Z","timestamp":596937600000},"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":[[1988,12]]},"DOI":"10.1007\/bf02187915","type":"journal-article","created":{"date-parts":[[2005,10,29]],"date-time":"2005-10-29T08:11:50Z","timestamp":1130573510000},"page":"307-324","source":"Crossref","is-referenced-by-count":24,"title":["Valuations and polarity"],"prefix":"10.1007","volume":"3","author":[{"given":"Jim","family":"Lawrence","sequence":"first","affiliation":[]}],"member":"297","published-online":{"date-parts":[[1988,12,1]]},"reference":[{"key":"BF02187915_CR1","unstructured":"G. Birkhoff,Lattice Theory, Third edn., American Mathematical Society, Providence, RI, 1979."},{"key":"BF02187915_CR2","doi-asserted-by":"crossref","first-page":"230","DOI":"10.1007\/BF01228204","volume":"24","author":"L. Geissinger","year":"1973","unstructured":"L. Geissinger, Valuations on distributive lattices, I, II, III,Arch. Math. 24 (1973), 230\u2013239, 337\u2013345, 475\u2013481.","journal-title":"Arch. Math."},{"key":"BF02187915_CR3","doi-asserted-by":"crossref","first-page":"315","DOI":"10.1007\/BF01418228","volume":"211","author":"H. Groemer","year":"1974","unstructured":"H. Groemer, On the Euler characteristic in spaces with a separability property,Math. Ann. 211 (1974), 315\u2013321.","journal-title":"Math. Ann."},{"key":"BF02187915_CR4","doi-asserted-by":"crossref","first-page":"141","DOI":"10.1007\/BF00181456","volume":"6","author":"H. Groemer","year":"1977","unstructured":"H. Groemer, Minkowski addition and mixed volumes,Geom. Dedicat. 6 (1977), 141\u2013163.","journal-title":"Geom. Dedicat."},{"key":"BF02187915_CR5","doi-asserted-by":"crossref","first-page":"397","DOI":"10.2140\/pjm.1978.75.397","volume":"75","author":"H. Groemer","year":"1978","unstructured":"H. Groemer, On the extension of additive functionals on classes of convex sets,Pacific J. Math. 75 (1978), 397\u2013410.","journal-title":"Pacific J. Math."},{"key":"BF02187915_CR6","unstructured":"H. Groemer, On rings of sets associated with characteristic functions, preprint."},{"key":"BF02187915_CR7","volume-title":"Convex Polytopes","author":"B. Gr\u00fcnbaum","year":"1967","unstructured":"B. Gr\u00fcnbaum,Convex Polytopes, Interscience, New York, 1967."},{"key":"BF02187915_CR8","doi-asserted-by":"crossref","first-page":"101","DOI":"10.1515\/crll.1955.194.101","volume":"194","author":"H. Hadwiger","year":"1955","unstructured":"H. Hadwiger, Euler charackteristik und kombinatorische geometrie,J. Reine Angew. Math. 194 (1955), 101\u2013110.","journal-title":"J. Reine Angew. Math."},{"key":"BF02187915_CR9","doi-asserted-by":"crossref","first-page":"103","DOI":"10.1215\/S0012-7094-55-02209-2","volume":"22","author":"P. Hammer","year":"1955","unstructured":"P. Hammer, Maximal convex sets,Duke Math. J. 22 (1955), 103\u2013106.","journal-title":"Duke Math. J."},{"key":"BF02187915_CR10","unstructured":"R. Jamison, A general duality between the theorems of Caratheodory and Helly, to appear."},{"key":"BF02187915_CR11","doi-asserted-by":"crossref","first-page":"119","DOI":"10.2307\/2312880","volume":"70","author":"V. Klee","year":"1963","unstructured":"V. Klee, The Euler characteristic in combinatorial geometry,Amer. Math. Monthly 70 (1963), 119\u2013127.","journal-title":"Amer. Math. Monthly"},{"key":"BF02187915_CR12","unstructured":"J. Lawrence, Clustered families of convex sets,Houston J. Math., to appear."},{"key":"BF02187915_CR13","unstructured":"J. Lawrence, Minkowski rings, to appear."},{"key":"BF02187915_CR14","doi-asserted-by":"crossref","first-page":"170","DOI":"10.1007\/978-3-0348-5858-8_9","volume-title":"Convexity and its applications","author":"P. McMullen","year":"1983","unstructured":"P. McMullen and R. Schneider, Valuations on convex bodies, inConvexity and its applications (P. M. Gruber and J. M. Wills, eds.), 170\u2013247, Birkhauser, Basel, 1983."},{"key":"BF02187915_CR15","volume-title":"Convex Analysis","author":"R. T. Rockafellar","year":"1972","unstructured":"R. T. Rockafellar,Convex Analysis, Princeton University Press, Princeton, NJ, 1972."},{"key":"BF02187915_CR16","volume-title":"Studies in Pure Mathematics","author":"G.-C. Rota","year":"1971","unstructured":"G.-C. Rota, The valuation ring, inStudies in Pure Mathematics (L. Mirsky, ed.), Academic Press, New York, 1971."},{"key":"BF02187915_CR17","doi-asserted-by":"crossref","first-page":"1412","DOI":"10.4153\/CJM-1968-142-0","volume":"20","author":"G. T. Sallee","year":"1968","unstructured":"G. T. Sallee, Polytopes, valuations, and the Euler relation,Canad. J. Math. 20 (1968), 1412\u20131424.","journal-title":"Canad. J. Math."},{"key":"BF02187915_CR18","doi-asserted-by":"crossref","first-page":"597","DOI":"10.1112\/plms\/s3-18.4.597","volume":"18","author":"G. C. Shephard","year":"1968","unstructured":"G. C. Shephard, Euler-type relations for convex polytopes,Proc. London Math. Soc. 18 (1968), 597\u2013606.","journal-title":"Proc. London Math. Soc."},{"key":"BF02187915_CR19","volume-title":"Convexity and Optimization","author":"J. Stoer","year":"1970","unstructured":"J. Stoer and C. Witzgall,Convexity and Optimization, Springer-Verlag, New York, 1970."}],"container-title":["Discrete &amp; Computational Geometry"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02187915.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/article\/10.1007\/BF02187915\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/BF02187915","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,5,5]],"date-time":"2023-05-05T07:05:28Z","timestamp":1683270328000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/BF02187915"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[1988,12]]},"references-count":19,"journal-issue":{"issue":"4","published-print":{"date-parts":[[1988,12]]}},"alternative-id":["BF02187915"],"URL":"https:\/\/doi.org\/10.1007\/bf02187915","relation":{},"ISSN":["0179-5376","1432-0444"],"issn-type":[{"value":"0179-5376","type":"print"},{"value":"1432-0444","type":"electronic"}],"subject":[],"published":{"date-parts":[[1988,12]]}}}