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Let $X_1, ..., X_{n+1} \in \mathbb{R}^d$, $n > d$, be iid random variables following some distribution $F$ on $\mathbb{R}^d$. What is (a lower bound on) the probability that $X_{n+1}$ falls inside the (random) convex hull of $X_1, ..., X_n$?

I found solutions to a similar problem in Hayakawa, Lyons, and Oberhauser (2023), Theorem 14 (https://doi.org/10.1007/s00440-022-01186-1). They define $p_{n,X}(\theta)$ as the probability that some given vector $\theta \in \mathbb{R}^d$ (w.l.o.g. the origin, $\theta = 0$) falls inside the convex hull of $X_1, ..., X_n$, and derive a bound depending (only) on $n$ and $d$ (as well as some additional assumption on $F$). I, meanwhile, am interested in $p_{n,X}(X)$.

Can $p_{n,X}(\theta)$ be used to inform $p_{n,X}(X)$? What if we had distributional assumptions on $F$ (e.g. Gaussianity)?

Koechi
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    Welcome on MSE. Very nice first question (+1). Not a precise answer, just a simple remark that you may have made so far: if $d=1$, and $F$ admits a density with respect to the Lebesgue measure, then almost surely ${X_1,\dots,X_{n+1}}$ has cardinal $n+1$, and the only permutation $\sigma\in\mathfrak{S}{n+1}$ such that $X{\sigma(i)}<X_{\sigma(j)}$ whenever $i<j$ is uniformly distribution on the symmetric group. Therefore $\mathbb P(\min_{1\leqslant k\leqslant n} {X_k}<X_{n+1}<\max_{1\leqslant k\leqslant n} {X_k})=\mathbb P(\sigma(n+1)\notin{1,n+1})=\frac{n-1}{n+1}$. – Kolakoski54 Dec 10 '24 at 22:08
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    If F is uniformly distributed in the d-1 dimensional sphere, then the probability is 0, for any n. So some assumption on F is needed. – izzyg Dec 11 '24 at 08:27

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