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A multifacility location problem on median spaces
 Discrete Appl. Math
, 1996
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Medians of discrete sets according to a linear distance
, 2006
"... In this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set alo ..."
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In this paper, we present some results concerning the median points of a discrete set according to a distance defined by means of two directions p and q. We describe a local characterization of the median points and show how these points can be determined from the projections of the discrete set along directions p and q. We prove that the discrete sets having some connectivity properties have at most four median points according to a linear distance, and if there are four median points they form a parallelogram. Finally, we show that the 4connected sets which are convex along the diagonal directions contain their median points along these directions.
Pareto envelopes in simple polygons
, 2008
"... For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t ′ ∈ T such that d(x, t ′ ) < d(y, t ′). The set of nondominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in ..."
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For a set T of n points in a metric space (X, d), a point y ∈ X is dominated by a point x ∈ X if d(x, t) ≤ d(y, t) for all t ∈ T and there exists t ′ ∈ T such that d(x, t ′ ) < d(y, t ′). The set of nondominated points of X is called the Pareto envelope of T. H. Kuhn (1973) established that in Euclidean spaces, the Pareto envelopes and the convex hulls coincide. Chalmet et al. (1981) characterized the Pareto envelopes in the rectilinear plane (R2, d1) and constructed them in O(n log n) time. In this note, we investigate the Pareto envelopes of pointsets in simple polygons P endowed with geodesic d2 or d1metrics (i.e., Euclidean and Manhattan metrics). We show that Kuhn’s characterization extends to Pareto envelopes in simple polygons with d2metric, while that of Chalmet et al. extends to simple rectilinear polygons with d1metric. These characterizations provide efficient algorithms for construction of these Pareto envelopes.
COLLAPSIBLE POLYHEDRA AND MEDIAN SPACES
, 1998
"... It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsib ..."
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It is shown that a collapsible, compact, connected, simplicial polyhedron admits a cubical subdivision and a median convexity, such that all cubes are convex subspaces with a convexity of subcubes. Conversely, a compact, connected, cubical polyhedron with a convexity as described admits a collapsible simplicial subdivision. Such a convexity, when it exists, is uniquely determined by the corresponding cubical presentation. Some related open problems have been formulated. .
linear distance. Discrete and Computational Geometry, Springer Verlag (Germany), 2000, 23,
"... HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte p ..."
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HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et a ̀ la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés. cc sd