TITLE: Estimate probability( 0 is in the convex hull of N random points ) ?
QUESTION [5 upvotes]: Can anyone estimate N such that Prob( 0 is in the convex hull of $N$ points ) >= .95
for points uniformly scatterered in $[-1,1]^d$, $d = 2, 3, 4, 10$ ?
The application is nearest-neghbour interpolation:
given values $z_j$ at sample points $X_j$, and a query point $P$,
one chooses the $N$ $X_j$ nearest to $P$ ($N$ fixed) and averages their $z_j$.
If $P$ is not in the convex hull of the $N$ $X_j$,
the interpolation will be one-sided, not so good.
I'd like to be able to say
"taking 6 neighbors in 2d, 10 in 3d, is seldom one-sided".
If anyone could point me to selfcontained pseudocode for the function Inhull( $N$ points )
(without calling full LP), that would be useful too.
(Please add tags interpolation convex-geometry ?)

REPLY [17 votes]: This is a classical and essentially geometric problem. In fact, the answer does not depend 
on the distribution of the points (as long as the distribution is centrally symmetric). 
The following result is due to Wendel (link).

Theorem. If $X_1$, ..., $X_N$ are i.i.d. random points in $R^d$ whose distribution
  is symmetric with respect to $0$ and assigns measure zero to every hyperplane
  through $0$, then
  $$\mathbb P(0\notin \mbox{conv}\{X_1,\dots,X_N\})=\frac{1}{2^{N-1}}\sum\limits_{k=0}^{d-1}{N-1 \choose  k}.$$

The proof is straightforward. Let $\mu$ be the distribution of $X_k$, and set
$$ f(x_1,\dots,x_N) = \begin{cases} 1, & \mbox{if } x_1,\dots,x_N\ \mbox{ lie in an open halfspace of $\mathbb R^d$ with $0$ in the boundary}, \newline 0, & \mbox{else.} \end{cases}$$
Then due to the invariance of $\mu$ under reflection in the origin, we have that
$$\mathbb P(0\notin \mbox{conv}\{X_1,\dots,X_N\})=\int_{\mathbb R^d}\dots \int_{\mathbb R^d} \frac{1}{2^N}\sum\limits_{\varepsilon_i=\pm1}f(\varepsilon_1x_1,\dots,\varepsilon_Nx_N)\ \mu(dx_1)\dots\mu(dx_N).$$
Now, the sum
$$C(N,d)=\sum\limits_{\varepsilon_i=\pm1}f(\varepsilon_1x_1,\dots,\varepsilon_Nx_N)$$
can be interpreted as the number of connected components of the set $\mathbb R^d\backslash (H_1\cup\dots\cup H_N)$  induced by the hyperplanes $H_1$, ..., $H_N$ through $0$ which are in general position. But there is a classical calculation going back to to Steiner and Schläfli, which shows that
$$C(N,d)= 2\sum\limits_{k=0}^{d-1}{N-1 \choose  k}.$$