McMorris, F. R.; Mulder, Henry Martyn; Ortega, O. The \(\ell_{p}\)-function on trees. (English) Zbl 1251.90249 Networks 60, No. 2, 94-102 (2012). Summary: A \(p\)-value of a sequence \(\pi = (x_{1}, x_{2},\ldots , x_{k})\) of elements of a finite metric space \((X, d)\) is an element \(x\) for which \(\sum_{i=1}^k d^p (x,x_i)\) is minimal. The function \(\ell _{p}\) with domain the set of all finite sequences defined by \(\ell _{p}(\pi ):= \{x: x \text{ is a } p-\text{value of }\pi \}\) is called the \(\ell _{p}\)-function on \(X\). The \(\ell _{p}\)-functions with \(p = 1\) and \(p = 2\) are the well-studied median and mean functions respectively. In this article, the \(\ell _{p}\)-function on finite trees is characterized axiomatically. Cited in 8 Documents MSC: 90B80 Discrete location and assignment 05C82 Small world graphs, complex networks (graph-theoretic aspects) 05C05 Trees 90B30 Production models Keywords:location function; \(\ell_{p}\)-function; median function; mean function; tree PDFBibTeX XMLCite \textit{F. R. McMorris} et al., Networks 60, No. 2, 94--102 (2012; Zbl 1251.90249) Full Text: DOI References: [1] Handbook of social choice and welfare 1 (2002) · Zbl 1307.91009 [2] K. J. Arrow A. K. Sen K. Suzumura Handbook of social choice and welfare 2 Elsevier Amsterdam 2005 [3] Barthélemy, A formal theory of consensus, SIAM J Discr Math 4 pp 305– (1991) · Zbl 0734.90008 [4] Barthélemy, The median procedure in cluster analysis and social choice theory, Math Soc Sci 1 pp 235– (1981) · Zbl 0486.62057 [5] J. E. Biagi Facility location and the mean function 2000 [6] Day, Axiomatic consensus theory in group choice and biomathematics, Frontiers in Appl Math (2003) · Zbl 1031.92001 [7] Foster, An axiomatic characterization of a class of locations in tree networks, Oper Res 46 pp 347– (1998) · Zbl 0979.90118 [8] Holzman, An axiomatic approach to location on networks, Math Oper Res 15 pp 553– (1990) · Zbl 0715.90070 [9] McMorris, Axiomatic characterization of the mean function on trees, Discr Math Algor Appl 2 pp 313– (2010) · Zbl 1226.05087 [10] McMorris, The median function on distributive semilattices, Discr Appl Math 127 pp 319– (2003) · Zbl 1026.06007 [11] McMorris, The median procedure on median graphs, Discr Appl Math 84 pp 165– (1998) · Zbl 0906.05023 [12] McMorris, The center function on trees, Networks 38 pp 84– (2001) · Zbl 0990.90063 [13] Mulder, An axiomatization of the median procedure on the n-cube, Discr Appl Math 159 pp 939– (2011) · Zbl 1222.05245 [14] Mulder, Axiomization of the center function on trees, Australasian J Comb 41 pp 223– (2008) · Zbl 1145.05304 [15] Vohra, An axiomatic characterization of some locations in trees, Eur J Oper Res 90 pp 78– (1996) · Zbl 0914.90182 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.