Erdős distance problems in normed spaces

“…Exact values and extremal sets for up to 14 points ( Fig. 1) were determined by Schade [15] (see also [2]). Also many related problems 356 P. Brass were discussed, e.g., the maximum number of unit distances in other metric spaces, higher dimensions, convex or general positions, of smallest or largest distances, etc.…”

Section: Introductionmentioning

confidence: 99%

On Point Sets with Many Unit Distances in Few Directions

Braß

1998

Discrete Comput Geom

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We study the problem of the maximum number of unit distances among n points in the plane, under the additional restriction that we count only those unit distances that occur in a fixed set of k directions, taking the maximum over all sets of n points and all sets of k directions. We prove that, for fixed k and sufficiently large n > n 0 (k), the extremal sets are essentially sections of lattices, bounded by edges parallel to the k directions and of equal length.

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“…Recently the same but much harder question has been answered for packings of congruent circles in the hyperbolic plane by Bowen and Radin in the elegant papers [5,6]. Moreover, with the help of the properly modified method of [10] Brass [7] could prove that the maximum number of touching pairs in a packing of n translates of a convex body K in E 2 is N3n −`12n − 3M, if K is not a parallelogram, and …”

mentioning

confidence: 95%

On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body

Bezdek

2002

Journal of Combinatorial Theory, Series A

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Erdős distance problems in normed spaces

Cited by 30 publications

References 7 publications

Independence Numbers of Planar Contact Graphs

Independence Numbers of Planar Contact Graphs

On Point Sets with Many Unit Distances in Few Directions

On the Maximum Number of Touching Pairs in a Finite Packing of Translates of a Convex Body

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