This article is cited in 3 papers

Two New Approaches to Obtaining Estimates in the Danzer–Grünbaum Problem

L. V. Buchok

M. V. Lomonosov Moscow State University

Abstract: We use probabilistic methods to estimate the cardinality of a set $S$ in a Euclidean space such that no three points of $S$ form a right or an obtuse angle. Let $a(n)$ be the cardinality of a maximal subset $S\subset\mathbb R^n$ with this property. We prove that
$$ a(n)\ge\frac23\biggl\lfloor\sqrt2\biggl(\frac2{\sqrt3}\biggr)^n\biggr\rfloor. $$


Keywords: Euclidean space, angle, set of points, Danzer–Grünbaum problem, Erdős–Füredi method.

UDC: 514.11

Received: 29.12.2008

DOI: 10.4213/mzm6610

 English version: Mathematical Notes, 2010, 87:4, 489–496

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