iarc6io

2022-07-18

Non-zero probability of hitting a convex hull of $d+1$ i.i.d. points in ${\mathbb{R}}^{d}$.
Let ${X}_{1},\dots ,{X}_{d+1}$ be $d+1$ i.i.d. random points in ${\mathbb{R}}^{d}$ sampled from a continuous probability μ of density f.
Let ${x}_{0}\in {\mathbb{R}}^{d}$. Is it true that almost surely with respect to $\mu$, $\mathbb{P}\left({x}_{0}\in \mathrm{C}\mathrm{o}\mathrm{n}\mathrm{v}\left({X}_{1},\dots ,{X}_{d+1}\right)\right)>0$ where Conv denotes the convex hull of the points?

Alanna Downs

Expert

Step 1
Choose $\left(d+1\right)$ balls ${\mathsf{B}}_{1},\dots ,{\mathsf{B}}_{d+1}$ in ${\mathbb{R}}^{d}$ such that $0\in \mathrm{Conv}\left({x}_{1},\dots ,{x}_{d+1}\right)$ for any choices of ${x}_{i}\in {\mathsf{B}}_{i}$ for $i=1,\dots ,d+1$. For instance, fix a regular d-simplex centered at 0 and replace each vertex with a small ball.
Step 2
Now let ${x}_{0}$ be a Lebesgue point of f such that $f\left({x}_{0}\right)>0$. Then it is straightforward to prove that
$\underset{r\to {0}^{+}}{lim}\frac{1}{{r}^{d}|{\mathsf{B}}_{i}|}{\int }_{{x}_{0}+r{\mathsf{B}}_{i}}f\left(x\right)\phantom{\rule{thinmathspace}{0ex}}\mathrm{d}x=f\left({x}_{0}\right)>0$
for each $i=1,\dots ,d+1$. So if $r>0$ is sufficiently small, then
$\mathbb{P}\left({x}_{0}\in \mathrm{Conv}\left({X}_{1},\dots ,{X}_{d+1}\right)\right)\ge \mathbb{P}\left({\cap }_{i=1}^{d+1}\left\{{X}_{i}\in {x}_{0}+r{\mathsf{B}}_{i}\right\}\right)>0.$

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