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](https://en.wikipedia.org/wiki/Thomson_problem#Configurations_of_smallest_known_energy) 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.



Known configurations maximizing the volume of the convex hull of n points on the unit sphere
The problem is elementary for $n=5$.


We may regard that case as a combination of two triangular pyramids sharing a base $\triangle ABC$. Then the volume is always bounded by one-third the area of the common base times the distance between the two remaining points $D,E$ (the latter is always greater than or equal to the sum of the pyramid altitudes). Then this bound is saturated and both factors maximized by placing $D$ and $E$ at opposite points ("poles") and distributing $A,B,C$ uniformly along the "equator" (a maximally symmetric triangular bipyramid).




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Let's look more broadly here, as other broader answers have yet to appear.


We begin with [this 2014 answer from Math Stack Exchange](https://math.stackexchange.com/questions/979660/largest-n-vertex-polyhedron-that-fits-into-a-unit-sphere#988708), which concentrates on four to eight points in three dimensions. Summarizing those results gives:


$n=4$ -- regular tetrahedron


$n=5$ -- triangular bipyramid, as above


$n=6$ -- regular octahedron


$n=7$ -- pentagonal bipyramid (this would seem to overcrowd the "equator" with five points, but plausible alternatives such as the capped octahedron would be worse)


$n=8$ -- not a cube, which isn't even close; rather a $D\_2$-symmetry object with triangular faces and vertices of order $4$ and $5$.


The linked answer cites Ref. [1] for proof.


Horváth and Lángi[2] give a lemma that all solutions with four or more points in three dimensions are polyhedra with triangular faces; and the faces are also simplices in higher dimensions. The above claims conform with this, as do well-known (but AFAIK not rigorously proved) arrangements for larger $n$ such as the laterally tricapped trigonal prism for $n=9$ and the regular icosahedron for $n=12$. The triangular face requirement excludes the regular dodecahedron, which in any event has long since been beaten for $20$ points.


Reference [2] also considers higher dimensionalities. For $n=d+1$ points on the surface of a $d$-dinensional ball the optimum is the regular simplex. With $n=d+2$ the optimum is described as combining regular-simplex cross-sections in two complementary subsets of the dimensions. Going back to the triangular bipyramid with $n=5$ and $d=3$, the relevant cross-sections are an equilateral triangle (the base of the bipyramid) in two of the dimensions, and a line segment (the axis) in the remaining dimension. With higher dimensionality the two orthogonal cross-sections should have as nearly equal dimensionalities as possible. Thus for $(n,d)=(6,4)$ we would select an equilateral triangle in two dimensions and another equilateral triangle in the remaining two dimensions, not a three-dimensional tetrahedron and a one-dimensional line segment; so the $(6,4)$ optimum is not a higher-dimensional bipyramid. For $n=d+3$ with $d$ odd the optimum is guaranteed by using three orthogonal sim-plectic cross-sections, such as the line segments which are diagonals of the regular octahedron for $n=6, d=3$.


**References**


1. Joel D. Berman, Kit Hanes (1970), "Volumes of polyhedra inscribed in the unit sphere in E³",
*Mathematische Annalen*, **188**, 1, 78-84.
2. Horváth, Ákos G.; Lángi, Zsolt (2016), "Maximum volume polytopes inscribed in the unit sphere", *Monatsh. Math.* **181**, No. 2, 341-354. ZBL1354.52016.


