scholarly journals Some inequalities for planar convex sets containing one lattice point

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.

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

1999 ◽  
Vol 59 (1) ◽  
pp. 147-152 ◽  
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.

scholarly journals New inequalities for planar convex sets with lattice point constraints

1996 ◽  
Vol 54 (3) ◽  
pp. 391-396 ◽  
Author(s):  
Poh W. Awyong ◽  
Paul R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Integral Lattice ◽  
Planar Convex ◽  
New Inequalities

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.

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

1996 ◽  
Vol 53 (3) ◽  
pp. 469-478 ◽  
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.

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

1995 ◽  
Vol 52 (1) ◽  
pp. 137-151 ◽  
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.

scholarly journals Further inequalities for convex sets with lattice point constraints in the plane

1980 ◽  
Vol 21 (1) ◽  
pp. 7-12 ◽  
Author(s):  
P.R. Scott
Keyword(s):  
Lattice Point ◽  
Convex Sets ◽  
Convex Set ◽  
Integral Lattice ◽  
Image Position ◽  
Closed Convex Set

Let K be a bounded closed convex set in the plane containing no points of the integral lattice in its interior and having width w, area A, perimeter p and circumradius R. The following best possible inequalities are established:

scholarly journals On the area of planar convex sets containing many lattice points

1987 ◽  
Vol 35 (3) ◽  
pp. 441-454
Author(s):  
P. R. Scott
Keyword(s):  
Large Class ◽  
Convex Sets ◽  
Convex Set ◽  
Lattice Points ◽  
Symmetric Convex ◽  
Classical Theorem ◽  
Planar Convex

A classical theorem of van der Corput gives a bound for the volume of a symmetric convex set in terms of the number of lattice points it contains. This theorem is here generalized and extended for a large class of non-symmetric sets in the plane.

Two Inequalities for Planar Convex Sets

1985 ◽  
Vol 28 (1) ◽  
pp. 60-66 ◽  
Author(s):  
George Tsintsifas
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Compact Convex ◽  
Compact Convex Set ◽  
Planar Convex ◽  
Straight Lines

AbstractB. Grünbaum, J. N. Lillington and lately R. J. Gardner, S. Kwapien and D. P. Laurie have considered inequalities defined by three concurrent straight lines in the interior of a planar compact convex set. In this note we prove two elegant conjectures by R. J. Gardner, S. Kwapien and D. P. Laurie.

Applications of outwardly simple line families to plane convex sets

1977 ◽  
Vol 82 (3) ◽  
pp. 353-356 ◽  
Author(s):  
K. J. Falconer
Keyword(s):  
Convex Sets ◽  
Convex Set ◽  
Plane Convex ◽  
Minimal Width ◽  
Parallel Lines

We use the concept of outwardly simple line families (see Hammer and Sobczyk (4)) first to obtain conditions involving mid-chords that ensure that a plane convex set is centro-symmetric, and secondly to show that it is possible to inscribe a semicircle of diameter ω in any convex set of minimal width ω in at least 3 different ways. We show that a plane convex set X is centro-symmetric if every mid-chord of X (that is every chord of X mid-way between two parallel lines of support of X) bisects the area of X, or alternatively if every mid-chord of X is a diameter of X (that is a longest chord in some direction). Hammer and Smith (3) have used outwardly simple line families in a different way to show that a plane convex set X is centro-symmetric if every diameter of X bisects the area of X, or if every diameter of X bisects the perimeter of X.

scholarly journals Maximizing the overlap of two planar convex sets under rigid motions

2007 ◽  
Vol 37 (1) ◽  
pp. 3-15 ◽  
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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