Geodesic
Quantifying minimal noncollinearity among random points
Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 4, pp. 753-768 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $\varphi_ {n, K} $ denote the largest angle in all the triangles with vertices among the $ n $ points selected at random in a compact convex subset $ K $ of $\mathbb {R}^ d $ with nonempty interior, where $ d\ge2 $. It is shown that the distribution of the random variable $\lambda_d (K)\,\frac {n^ 3}{3!}\,(\pi-\varphi_ {n, K})^{d-1} $, where $\lambda_d (K) $ is a certain positive real number which depends only on the dimension $d$ and the shape of $K$, converges to the standard exponential distribution as $n\to\infty$. By using the Steiner symmetrization, it is also shown that $\lambda_d (K)$, which is referred to in the paper as the elongation of $K$, attains its minimum if and only if $K$ is a ball $B^{(d)}$ in $\mathbf {R}^d$. Finally, the asymptotics of $\lambda_d(B^{(d)})$ for large $d$ is determined.
Keywords: convex sets, random points, geometric probability theory, integral geometry, Steiner symmetrization, asymptotic approximation.
Mots-clés : maximal angle, convergence in distribution 
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     author = {I. Pinelis},
     title = {Quantifying minimal noncollinearity among random points},
     journal = {Teori\^a vero\^atnostej i ee primeneni\^a},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TVP_2017_62_4_a5/}
}
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I. Pinelis. Quantifying minimal noncollinearity among random points. Teoriâ veroâtnostej i ee primeneniâ, Tome 62 (2017) no. 4, pp. 753-768. http://geodesic.mathdoc.fr/item/TVP_2017_62_4_a5/

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