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Gro-Tsen
Gro-Tsen
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For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the problem of computing $V_n$ has been extensively studied and has a standard name: what is this name?

For which values of $n$ is the exact value of $V_n$, and/or a configuration realizing it, known (at least combinatorially)? And, for small $n$, what are the best known configurations even if they are not proved to be optimal? Essentially, I'm looking for a list analogous to what Wikipedia lists for the Thomson problem but for the volume of the convex hull instead of electrostatic potential energy.

PS: the dual problem of finding the least possible volume cut by $n$ (hyper)planes all tangent to the unit sphere (i.e., the smallest convex polytope exscribed around a sphere, rather than the largest inscribed in one as described above) also seems interesting, so if it has a standard name I'm also curious to know.

For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the problem of computing $V_n$ has been extensively studied and has a standard name: what is this name?

For which values of $n$ is the exact value of $V_n$, and/or a configuration realizing it, known (at least combinatorially)? And, for small $n$, what are the best known configurations even if they are not proved to be optimal? Essentially, I'm looking for a list analogous to what Wikipedia lists for the Thomson problem but for the volume of the convex hull instead of electrostatic potential energy.

For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the problem of computing $V_n$ has been extensively studied and has a standard name: what is this name?

For which values of $n$ is the exact value of $V_n$, and/or a configuration realizing it, known (at least combinatorially)? And, for small $n$, what are the best known configurations even if they are not proved to be optimal? Essentially, I'm looking for a list analogous to what Wikipedia lists for the Thomson problem but for the volume of the convex hull instead of electrostatic potential energy.

PS: the dual problem of finding the least possible volume cut by $n$ (hyper)planes all tangent to the unit sphere (i.e., the smallest convex polytope exscribed around a sphere, rather than the largest inscribed in one as described above) also seems interesting, so if it has a standard name I'm also curious to know.

Source Link
Gro-Tsen
Gro-Tsen
  • 32.4k
  • 5
  • 86
  • 372

Known configurations maximizing the volume of the convex hull of n points on the unit sphere

For $n\geq 4$, let $V_n$ be the maximum volume of the convex hull of $n$ points on the unit sphere (in $\mathbb{R}^3$, although information on higher dimensions is welcome as well). I'm sure the problem of computing $V_n$ has been extensively studied and has a standard name: what is this name?

For which values of $n$ is the exact value of $V_n$, and/or a configuration realizing it, known (at least combinatorially)? And, for small $n$, what are the best known configurations even if they are not proved to be optimal? Essentially, I'm looking for a list analogous to what Wikipedia lists for the Thomson problem but for the volume of the convex hull instead of electrostatic potential energy.