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- W2950594920 abstract "Let $S$ be a $k$-colored (finite) set of $n$ points in $mathbb{R}^d$, $dgeq 3$, in general position, that is, no {$(d + 1)$} points of $S$ lie in a common $(d - 1)$}-dimensional hyperplane. We count the number of empty monochromatic $d$-simplices determined by $S$, that is, simplices which have only points from one color class of $S$ as vertices and no points of $S$ in their interior. For $3 leq k leq d$ we provide a lower bound of $Omega(n^{d-k+1+2^{-d}})$ and strengthen this to $Omega(n^{d-2/3})$ for $k=2$. On the way we provide various results on triangulations of point sets in $mathbb{R}^d$. In particular, for any constant dimension $dgeq3$, we prove that every set of $n$ points ($n$ sufficiently large), in general position in $mathbb{R}^d$, admits a triangulation with at least $dn+Omega(log n)$ simplices." @default.
- W2950594920 created "2019-06-27" @default.
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- W2950594920 date "2012-10-26" @default.
- W2950594920 modified "2023-10-01" @default.
- W2950594920 title "Empty Monochromatic Simplices" @default.
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