ON PLANAR CONVEX SETS CONTAINING ONE LATTICE POINT

We obtain new inequalities relating the inradius of a planar convex set with interior containing no point of the integral lattice, with the area, perimeter and diameter of the set. By considering a special sublattice of the integral lattice, we also obtain an inequality concerning the inradius and area of a planar convex set with interior containing exactly one point of the integral lattice.

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Some inequalities for planar convex sets containing one lattice point

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032093 ◽

1998 ◽

Vol 58 (1) ◽

pp. 159-166

Author(s):

M. A. Hernández Cifre ◽

S. Segura Gomis

Keyword(s):

Lattice Point ◽

Convex Sets ◽

Convex Set ◽

Integer Lattice ◽

Minimal Width ◽

Planar Convex

We obtain two inequalities relating the diameter and the (minimal) width with the area of a planar convex set containing exactly one point of the integer lattice in its interior. They are best possible. We then use these results to obtain some related inequalities.

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Circumradius-diameter and width-inradius relations for lattice constrained convex sets

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700032706 ◽

1999 ◽

Vol 59 (1) ◽

pp. 147-152 ◽

Cited By ~ 1

Author(s):

Poh Wah Awyong ◽

Paul R. Scott

Keyword(s):

Lattice Point ◽

Convex Sets ◽

Convex Set ◽

Compact Convex ◽

Integer Lattice ◽

Lattice Points ◽

Compact Convex Set ◽

Planar Convex ◽

Rectangular Lattices

Let K be a planar, compact, convex set with circumradius R, diameter d, width w and inradius r, and containing no points of the integer lattice. We generalise inequalities concerning the ‘dual’ quantities (2R − d) and (w − 2r) to rectangular lattices. We then use these results to obtain corresponding inequalities for a planar convex set with two interior lattice points. Finally, we conjecture corresponding results for sets containing one interior lattice point.

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Width-diameter relations for planar convex sets with lattice point constraints

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700017238 ◽

1996 ◽

Vol 53 (3) ◽

pp. 469-478 ◽

Cited By ~ 2

Author(s):

Poh W. Awyong ◽

Paul R. Scott

Keyword(s):

Lattice Point ◽

Convex Sets ◽

Convex Set ◽

Rectangular Lattice ◽

Integral Lattice ◽

Planar Convex

We obtain an inequality concerning the width and diameter of a planar convex set with interior containing no point of the rectangular lattice. We then use the result to obtain a corresponding inequality for a planar convex set with interior containing exactly two points of the integral lattice.

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Maximizing the overlap of two planar convex sets under rigid motions

Computational Geometry ◽

10.1016/j.comgeo.2006.01.005 ◽

2007 ◽

Vol 37 (1) ◽

pp. 3-15 ◽

Cited By ~ 14

Author(s):

Hee-Kap Ahn ◽

Otfried Cheong ◽

Chong-Dae Park ◽

Chan-Su Shin ◽

Antoine Vigneron

Keyword(s):

Convex Sets ◽

Planar Convex ◽

Rigid Motions

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Translations of planar convex sets

Archiv der Mathematik ◽

10.1007/bf01222547 ◽

1971 ◽

Vol 22 (1) ◽

pp. 103-105 ◽

Cited By ~ 4

Author(s):

Togo Nishiura ◽

Franz Schnitzer

Keyword(s):

Convex Sets ◽

Planar Convex

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On the maximal circumradius of a planar convex set containing one lattice point

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972700014519 ◽

1995 ◽

Vol 52 (1) ◽

pp. 137-151 ◽

Cited By ~ 1

Author(s):

Poh W. Awyong ◽

Paul R. Scott

Keyword(s):

Lattice Point ◽

Convex Set ◽

Compact Convex ◽

Lattice Points ◽

Compact Convex Set ◽

Planar Convex

We obtain a result about the maximal circumradius of a planar compact convex set having circumcentre O and containing no non-zero lattice points in its interior. In addition, we show that under certain conditions, the set with maximal circumradius is a triangle with an edge containing two lattice points.

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