1Universidade Federal do Ceará, Fortaleza, Brazil 2Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil 3Technische Universität Chemnitz, Chemnitz, Germany
Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 443-450
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Fabricio Siqueira Benevides; Carlos Hoppen; Hanno Lefmann; Knut Odermann; Fabricio Siqueira Benevides; Carlos Hoppen; Hanno Lefmann; Knut Odermann. On Heilbronn triangle-type problems in higher dimensions. Acta mathematica Universitatis Comenianae, Tome 88 (2019) no. 3, pp. 443-450. http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a13/
@article{AMUC_2019_88_3_a13,
author = {Fabricio Siqueira Benevides and Carlos Hoppen and Hanno Lefmann and Knut Odermann and Fabricio Siqueira Benevides and Carlos Hoppen and Hanno Lefmann and Knut Odermann},
title = { On {Heilbronn} triangle-type problems in higher dimensions},
journal = {Acta mathematica Universitatis Comenianae},
pages = {443--450},
year = {2019},
volume = {88},
number = {3},
url = {http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a13/}
}
TY - JOUR
AU - Fabricio Siqueira Benevides
AU - Carlos Hoppen
AU - Hanno Lefmann
AU - Knut Odermann
AU - Fabricio Siqueira Benevides
AU - Carlos Hoppen
AU - Hanno Lefmann
AU - Knut Odermann
TI - On Heilbronn triangle-type problems in higher dimensions
JO - Acta mathematica Universitatis Comenianae
PY - 2019
SP - 443
EP - 450
VL - 88
IS - 3
UR - http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a13/
ID - AMUC_2019_88_3_a13
ER -
%0 Journal Article
%A Fabricio Siqueira Benevides
%A Carlos Hoppen
%A Hanno Lefmann
%A Knut Odermann
%A Fabricio Siqueira Benevides
%A Carlos Hoppen
%A Hanno Lefmann
%A Knut Odermann
%T On Heilbronn triangle-type problems in higher dimensions
%J Acta mathematica Universitatis Comenianae
%D 2019
%P 443-450
%V 88
%N 3
%U http://geodesic.mathdoc.fr/item/AMUC_2019_88_3_a13/
%F AMUC_2019_88_3_a13
The Heilbronn triangle problem is a classical geometrical problem that asks for a placement of $n$ points in the unit-square $[0,1]^2$, that maximizes the smallest area of a triangle formed by those points. This problem has natural generalizations to higher dimensions. For integers $k, d \ge 2$ and a set $\mathcal P$ of $n$ points in $[0,1]^d$, let $s = \min\{(k-1),d\}$ and $V_k^{(d)}({\mathcal P})$ be the minimum $s$-dimensional volume of the convex hull of $k$ points in $\mathcal P$. Here, instead of considering the supremum of $V_k^{(d)}({\mathcal P})$, we consider its average value, $\avrg{\Delta}_k^{(d)}(n)$, when the $n$ points in $\mathcal P$ are chosen independently and uniformly at random in $[0,1]^d$. We prove that $\avrg{\Delta}_k^{(d)}(n) = \Theta \left(n^{\frac{-k}{1+|d-k+1|}}\right)$, for every fixed $k, d \ge 2$.
