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 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 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 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 On the width of a planar convex set containing zero or one lattice points

1993 ◽  
Vol 48 (1) ◽  
pp. 47-53
Author(s):  
Paul R. Scott
Keyword(s):  
Triangular Lattice ◽  
Short Proof ◽  
Convex Set ◽  
Compact Convex ◽  
Lattice Points ◽  
Rectangular Lattice ◽  
Integral Lattice ◽  
Compact Convex Set ◽  
Maximal Width ◽  
Planar Convex

We generalise to a rectangular lattice a known result about the maximal width of a planar compact convex set containing no points of the integral lattice. As a corollary we give a new short proof that the planar compact convex set of greatest width which contains just one point of the triangular lattice is an equilateral triangle.

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 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.

scholarly journals An isoperimetric problem with lattice point constraints

1979 ◽  
Vol 27 (1) ◽  
pp. 27-36 ◽  
Author(s):  
J. R. Arkinstall ◽  
P. R. Scott
Keyword(s):  
Lattice Point ◽  
Euclidean Plane ◽  
Convex Sets ◽  
Isoperimetric Problem ◽  
Zero Point ◽  
Integral Lattice ◽  
Bounded Convex

AbstractThe isoperimetric problem in the Euclidean plane is completely solved for bounded, convex sets which are symmetric about the origin, and which contain no non-zero point of the integral lattice.

Sign in / Sign up

Export Citation Format

Share Document