Two new approaches to obtaining estimates in the Danzer-Grünbaum problem

Abstract

We use probabilistic methods to estimate the cardinality of a set S in a Euclidean space such that no three points of S forma right or an obtuse angle. Let a(n) be the cardinality of a maximal subset S ⊂ ℜⁿ with this property. We prove that

$$ a\left( n \right) \geqslant \frac{2} {3}\left\lfloor {\sqrt 2 \left( {\frac{2} {{\sqrt 3 }}} \right)^n } \right\rfloor $$

.