So I imagined this problem back in 2017 or 2018.
Let $n>k\ge 0$ be integers.
Take a $n$-hypercube with side $1$.
Now on each "face" of $k$ dimensions, conviniently place a point.
Find $m$, where $m$ is the least dimension the convex hull of all the points can have. Find $V$, where $V$ is the least $m$-volume a convex hull of $m$ dimensions of all the points can have.
So far I have found:
$n=1, k=0 \implies m=1, V=1$
$n=2, k=0 \implies m=2, V=1$
$n=2, k=1 \implies m=1, V=\sqrt{2}$
$n=3, k=0 \implies m=3, V=1$
$n=3, k=1 \implies m=3, V=1/3$
$n=3, k=2 \implies m=1, V=\sqrt{3}$
$k=0 \implies m=n, V=1$
$k=n-1 \implies m=1, V=\sqrt{n}$
Does anyone know how to find more values of this?