On the structure of the set of integral points inside a ball

C. R. Math. Rep. Acad. Sci. Canada Vol. 33 (3) 2011, pp. 93–96
(Received: 2010-12-01 )

Konstantin Matveev, Department of Mathematics, University of Toronto; e-mail: kostya.matveev@utoronto.ca

Abstract/Résumé:

On the structure of the set of integral points inside a ball Resume/Abstract: We prove that for each open ball in \(\mathbb{R}^n\) of radius \(r \geq \frac{\sqrt{n+3}}{2}\), its center is contained in the convex hull of all integral points inside it. We also show that this estimate is sharp, i.e., for balls of radius \(r < \frac{\sqrt{n+3}}{2}\), the property does not hold.

Nous prouvons que, pour chaque balle ouverte dans \(\mathbb{R}^n\) de radius \(r \geq \frac{\sqrt{n+3}}{2}\), le centre se tient dans une enveloppe convexe de points à l’intérieur avec des coordonnées entières. Nous allons également montrer que cette estimation est forte, c’est-à-dire que la propriété ne tient pas pour les balles de radius \(r < \frac{\sqrt{n+3}}{2}\).

Keywords: convex hull, discrete geometry, integral points

AMS Subject Classification: Lattices and convex bodies in $n$ dimensions 52C07

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