zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Locating a central hunter on the plane. (English) Zbl 1146.90462
Summary: Protection, surveillance or other types of coverage services of mobile points call for different, asymmetric distance measures than the traditional Euclidean, rectangular or other norms used for fixed points. In this paper, the destinations are mobile points (prey) moving at fixed speeds and directions and the facility (hunter) can capture them using one of two possible strategies: either it is smart, predicting the prey’s movement in order to minimize the time needed to capture it, or it is dumb, following a pursuit curve, by moving at any moment in the direction of the prey. In either case, the hunter location in a plane is sought in order to minimize the maximum time of capture of any prey. An efficient solution algorithm is developed that uses the particular geometry that both versions of this problem possess. In the case of unpredictable movement of prey, a worst-case type solution is proposed, which reduces to the well-known weighted Euclidean minimax location problem.

90B85Continuous location
Full Text: DOI
[1] Plastria, F.: Continuous covering location problems. In: Hamacher, H., Drezner, Z. (eds.) Location Analysis: Theory and Applications, pp. 39--83. Springer, Berlin (2001)
[2] Barton, J.C., Eliezer, C.J.: Pursuit curves II. Bull. Inst. Math. Appl. 31, 139--141 (1995)
[3] Barton, J.C., Eliezer, C.J.: On pursuit curves. J. Aust. Math. Soc. 41, 358--371 (2000) · Zbl 0947.34055 · doi:10.1017/S0334270000011292
[4] Cera, M., Ortega, F.A.: Locating the median hunter among n mobile preys on the plane. Int. J. Ind. Eng.: Theory Appl. Pract. 9, 6--15 (2002)
[5] Plastria, F.: On destination optimality in asymmetric distance Fermat--Weber problems. Ann. Oper. Res. 40, 355--369 (1992) · Zbl 0783.90061 · doi:10.1007/BF02060487
[6] Pelegrín, B., Michelot, C., Plastria, F.: On the uniqueness of optimal solutions in continuous location theory. Eur. J. Operat. Res. 20, 327--331 (1985) · Zbl 0564.90012 · doi:10.1016/0377-2217(85)90005-0
[7] Drezner, Z.: On minmax optimization problems. Math. Program. 22, 227--230 (1982) · Zbl 0473.90067 · doi:10.1007/BF01581038
[8] Elzinga, D.J., Hearn, D.W.: Geometrical solutions for some minimax location problems. Transp. Sci. 6, 379--394 (1972) · doi:10.1287/trsc.6.4.379
[9] Charalambous, C.: Extension of the Elzinga--Hearn algorithm to the weighted case. Operat. Res. 30, 591--594 (1982) · Zbl 0484.90030 · doi:10.1287/opre.30.3.591
[10] Hearn, D.W., Vijay, J.: Efficient algorithms for the (weighted) minimum circle problem. Operat. Res. 30, 777--795 (1982) · Zbl 0486.90039 · doi:10.1287/opre.30.4.777
[11] Welzl, E.: Smallest enclosing disk (balls and ellipsoids), new results and new trends in computer science. In: Maurer, H. (ed.) Lecture Notes in Computer Science, vol. 555, pp. 359--370. Springer, New York (1991)
[12] Plastria, F.: Fully geometric solutions to some planar center minimax location problems. Stud. Locat. Analysis 7, 171--183 (1994) · Zbl 0891.90108
[13] Chrystal, G.: On the problem to construct the minimum circle enclosing n given points in the plane. Proc. Edinb. Math. Soc. 3, 30--33 (1885) · Zbl 17.0540.01 · doi:10.1017/S0013091500037238
[14] Shamos, M.I., Hoey, D.: Closest point problems. In: Proceedings of the 16th Annual Symposium on Foundations of Computer Science, pp. 151--162 (1975)
[15] Megiddo, N.: The weighted Euclidean 1--center problem. Math. Operat. Res. 8, 498--504 (1983) · Zbl 0533.90030 · doi:10.1287/moor.8.4.498