**
Beiträge zur Algebra und Geometrie **

Contributions to Algebra and Geometry

38(2), 423-427 (1997)

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On the Centre of Gravity and Width of Lattice-constrained Convex Sets in the Plane

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Maria de los Angeles Hernandez Cifre, P.R. Scott, Salvador Segura Gomis

1,3: Departamento de Matemáticas, Universidad de Murcia

30100-Murcia, Spain 2: Department of Pure Mathematics, University of Adelaide

South Australia 5005

mhcifre@fcu.um.es pscott@maths.adelaide.edu.au salsegom@fcu.um.es

**Abstract:** In 1982, Scott proposed the following conjecture: Let ${\bf Z}^2$ denote the lattice of integer points in the plane, and let $K$ be a convex set having the centroid as the only interior lattice point; then, $\omega (K) \leq {3 \sqrt{2}\over 2}$. In this paper we give a counterexample to that conjecture and propose another bound. Besides, we prove that $\omega (K) \leq {3 \sqrt{2}\over 2}$ for the family of all triangles.

**Classification (MSC91):** 52C05, 52A40

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