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Dorian
Dorian
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More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}>0$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. When $n=4$ as pointed out by the commenter below, the solution would be a tetrahedron. For higher $n$ the answer is not clear to me. I would also be interested in the patterns in the asymptotic limit as $n \to +\infty$.

I presume that if this is known, it is a well established result in graph theory, and I would appreciate any references.

Update: Thanks for the helpful comment and the answer. This answers my question.

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}>0$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. When $n=4$ as pointed out by the commenter below, the solution would be a tetrahedron. For higher $n$ the answer is not clear to me. I would also be interested in the patterns in the asymptotic limit as $n \to +\infty$.

I presume that if this is known, it is a well established result in graph theory, and I would appreciate any references.

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}>0$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. When $n=4$, the solution would be a tetrahedron. For higher $n$ the answer is not clear to me. I would also be interested in the patterns in the asymptotic limit as $n \to +\infty$.

I presume that if this is known, it is a well established result in graph theory, and I would appreciate any references.

Update: Thanks for the helpful comment and the answer. This answers my question.

Updated information obtained in comment.
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Dorian
Dorian
  • 2.6k
  • 23
  • 32

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}$$\min_{i \neq j} |a_i-a_j| = r_{min}>0$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. Already whenWhen $n=4$ as pointed out by the answer iscommenter below, the solution would be a not cleartetrahedron. For higher $n$ the answer is not clear to me. I would also be interested in the patterns in the asymptotic limit as $n \to +\infty$.

I presume that if this is known, it is a well established result in graph theory, and I would appreciate any references.

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. Already when $n=4$ the answer is not clear to me.

I presume that if this is known it is a well established result in graph theory and I would appreciate any references.

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}>0$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. When $n=4$ as pointed out by the commenter below, the solution would be a tetrahedron. For higher $n$ the answer is not clear to me. I would also be interested in the patterns in the asymptotic limit as $n \to +\infty$.

I presume that if this is known, it is a well established result in graph theory, and I would appreciate any references.

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Dorian
Dorian
  • 2.6k
  • 23
  • 32

How to minimize the length of a graph connecting n points in $\mathbb{R}^3$

More precisely I would like to consider the following problem:

Let $\{a_i\}_{i=1}^n$ be n points in $\mathbb{R}^3$ and assume I have the constraint $\min_{i \neq j} |a_i-a_j| = r_{min}$. How can I place $n$ points in $\mathbb{R}^3$ as to minimize $\sum_{i \neq j} |a_i-a_j|.$

When $n=2$ this is trivial. When $n=3$ the minimizer will clearly be an equilateral triangle with a point at each vertex. Already when $n=4$ the answer is not clear to me.

I presume that if this is known it is a well established result in graph theory and I would appreciate any references.