Construct certain pairs of $n$-dimensional convex sets.

Question

In $n$-dimensions, construct a convex set $A$--not a hypersphere--having an inscribed convex set $B$ of volume $p$ times that of $A$, for

$n=9$, $p =\frac{29}{64}$,

$n=15$, $p=\frac{8}{33}$,

$n=27$, $p=\frac{26}{323}$.

Answer 1

It has been formally proven LovasAndaiInvariance that the first case ($n=9$, $p=\frac{29}{64}$) can be obtained with the larger convex set $A$ being the "two-rebit density matrices", that is $4 \times 4$ symmetric, positive-definite matrices of trace 1. Further, $B$ is the subset of separable two-rebit density matrices--those that can be expressed as sums of Kronecker products of single-rebit density matrices, $2 \times 2$ symmetric, positive-definite matrices of trace 1. (The Euclidean [Hilbert-Schmidt] measure is employed.)

Though, not yet formally proven, the numerical and analytic evidence is very strong MasterLovasAndaithat the second case ($n=15$, $p=\frac{8}{33}$) is achieved with the larger convex set $A$ being the "two-qubit density matrices", that is $4 \times 4$ Hermitian, positive-definite matrices of trace 1. Further, $B$ is the subset of separable two-qubit density matrices--those that can be expressed as sums of Kronecker products of single-qubit density matrices, $2 \times 2$ Hermitian, positive-definite matrices of trace 1. (The Euclidean [Hilbert-Schmidt] measure is again employed.)

The third case ($n=27$, $p=\frac{26}{323}$) appears to be fulfilled (MasterLovasAndai) with the larger convex set $A$ being the "two-quater[nionic]bit density matrices", that is $4 \times 4$ Hermitian, positive-definite matrices of trace 1 with off-diagonal quaternionic entries . Further, $B$ is the subset of separable two-quaterbit density matrices--those that can be expressed as sums of Kronecker products of single-qubit density matrices, $2 \times 2$ Hermitian, positive-definite matrices of trace 1 with off-diagonal quaternionic entries. (The Euclidean [Hilbert-Schmidt] measure is employed as in the other two cases.)