Skip to main content
replaced http://mathoverflow.net/ with https://mathoverflow.net/
Source Link

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m,n\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional lineMean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cubeMean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m,n\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m,n\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

added 2 characters in body
Source Link
Minkov
  • 1.1k
  • 10
  • 23

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m\to\infty$$m,n\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m,n\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

deleted 7 characters in body
Source Link
Minkov
  • 1.1k
  • 10
  • 23

Let us considerConsider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

Let us consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

Consider $m$ points $v_1, \ldots, v_m \in R^{n}$, which are uniformly distributed on the $n$-dimensional unit sphere $S^{n-1} = \{v:\|v\|_2 = 1\}$. Let the minimum separation be $$ \rho = \min_{i,j\in{\{1,\ldots,m\}}} \|v_i - v_j\|_2. $$ Question: What is the expectation of $\rho$? How fast does $\rho$ converge to its expectation as $m\to\infty$?

Here are some closely related questions:

Mean minimum distance for N random points on a one-dimensional line

Mean minimum distance for N random points on a unit square (plane)

Mean minimum distance for K random points on a N-dimensional (hyper-)cube

However, the case for n-dimensional sphere seems less clear.

added 7 characters in body
Source Link
Minkov
  • 1.1k
  • 10
  • 23
Loading
edited title
Link
Minkov
  • 1.1k
  • 10
  • 23
Loading
added 4 characters in body
Source Link
Minkov
  • 1.1k
  • 10
  • 23
Loading
added 16 characters in body
Source Link
Minkov
  • 1.1k
  • 10
  • 23
Loading
added 379 characters in body
Source Link
Minkov
  • 1.1k
  • 10
  • 23
Loading
Source Link
Minkov
  • 1.1k
  • 10
  • 23
Loading