For i.i.d. points chosen in a bounded subset of $\mathbb{R}^n$ \mathbb{R}^3$ (or $\mathbb{R}^d$) it seems to me that $\theta_\min(n)\to 0$ is ensured when the support of the distribution has a smooth boundary. This covers the case of the uniform distribution on an Euclidean ball, and a uniform spherical distribution as well. (I'm not quite sure about how to state a converse).
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For i.i.d. points chosen in a bounded subset of $\mathbb{R}^n$ it seems to me that $\theta_\min(n)\to 0$ is equivalent to: the convex hull of ensured when the support of the distribution has a smooth boundary(I'll add details at request). This covers the case of the uniform distribution on an Euclidean ball, and the a uniform surface spherical distribution as well. (I'm not quite sure about how to state a converse). |
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For i.i.d. points chosen in a bounded subset of $\mathbb{R}^n$ it seems to me that $\theta_\min(n)\to 0$ is equivalent to: the convex hull of the support of the distribution has a smooth boundary (I'll add details at request). This covers the case of the uniform distribution and the uniform surface distribution as well. |
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