Diameter and radius in the Manhattan metric
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We investigate maximum size sets of lattice points with a given diameter,d, within a given rectilinearly bounded finite regionR inn dimensions, under the Manhattan orL1 metric. We show that when the length ofR in each dimension is an odd integer (as, for example, then-cube) there is, for every integerd, a maximum size set having radiusd/2 about some center, though the center need not be a lattice point.
Similar results are obtained whenR has even length in some dimensions, except for a set ofd values whose cardinality is one less than the number of dimensions in whichR has even length. This question is still open for these values.