Return to Answer

This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e_i$ and $v_j$ are the vertices of the unit balls in $\|\cdot\|_1$ and $\|\cdot\|_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e_i\|_\infty \le 1$, $\|T^{-1} v_j\|_1 \le s$ (or equivalently, $\|\text{Adj}(T) v_j\|_1 \le s |\det(T)|$, with $\det(T) \ne 0$).
Maple's Global Optimization Toolbox returns (almost immediately) an approximate solution, which is (up to roundoff error)

$$ T = \pmatrix{1/3 & 1 & 1\cr -1 & 1 & -1/3\cr 1 & 1/3 & -1\cr},\ s = \dfrac{9}{5}$$

It is easy to check that this does satisfy the constraints. Thus it appears that the answer is $9/5$. It should be possible to prove optimality rigoourously, if somewhat tediously, using the Karush-Kuhn-Tucker conditions.

EDIT: I also tried the $4$-dimensional case. The solution Maple obtained was

$$ T = \pmatrix{0 & 1 & -1 & 1\cr -1 & 0 & 1 & 1\cr 1 & -1 & 0 & 1\cr -1 & -1 & -1 & 0\cr}, \ s = 2 $$

This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e_i$ and $v_j$ are the vertices of the unit balls in $\|\cdot\|_1$ and $\|\cdot\|_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e_i\|_\infty \le 1$, $\|T^{-1} v_j\|_1 \le s$ (or equivalently, $\|\text{Adj}(T) v_j\|_1 \le s |\det(T)|$, with $\det(T) \ne 0$).
Maple's Global Optimization Toolbox returns (almost immediately) an approximate solution, which is (up to roundoff error)

$$ T = \pmatrix{1/3 & 1 & 1\cr -1 & 1 & -1/3\cr 1 & 1/3 & -1\cr},\ s = \dfrac{9}{5}$$

It is easy to check that this does satisfy the constraints. Thus it appears that the answer is $9/5$. It should be possible to prove optimality rigorously, if somewhat tediously, using the Karush-Kuhn-Tucker conditions.

EDIT: I also tried the $4$-dimensional case. The solution Maple obtained was

$$ T = \pmatrix{0 & 1 & -1 & 1\cr -1 & 0 & 1 & 1\cr 1 & -1 & 0 & 1\cr -1 & -1 & -1 & 0\cr}, \ s = 2 $$

This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e_i$ and $v_j$ are the vertices of the unit balls in $\|\cdot\|_1$ and $\|\cdot\|_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e_i\|_\infty \le 1$, $\|T^{-1} v_j\|_1 \le s$ (or equivalently, $\|\text{Adj}(T) v_j\|_1 \le s |\det(T)|$, with $\det(T) \ne 0$).
Maple's Global Optimization Toolbox returns (almost immediately) an approximate solution, which is (up to roundoff error)

$$ T = \pmatrix{1/3 & 1 & 1\cr -1 & 1 & -1/3\cr 1 & 1/3 & -1\cr},\ s = \dfrac{9}{5}$$

It is easy to check that this does satisfy the constraints. Thus it appears that the answer is $9/5$. It should be possible to prove optimality rigoourously, if somewhat tediously, using the Karush-Kuhn-Tucker conditions.

This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e_i$ and $v_j$ are the vertices of the unit balls in $\|\cdot\|_1$ and $\|\cdot\|_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e_i\|_\infty \le 1$, $\|T^{-1} v_j\|_1 \le s$ (or equivalently, $\|\text{Adj}(T) v_j\|_1 \le s |\det(T)|$, with $\det(T) \ne 0$).
Maple's Global Optimization Toolbox returns (almost immediately) an approximate solution, which is (up to roundoff error)

$$ T = \pmatrix{1/3 & 1 & 1\cr -1 & 1 & -1/3\cr 1 & 1/3 & -1\cr},\ s = \dfrac{9}{5}$$

It is easy to check that this does satisfy the constraints. Thus it appears that the answer is $9/5$. It should be possible to prove optimality rigoourously, if somewhat tediously, using the Karush-Kuhn-Tucker conditions.

EDIT: I also tried the $4$-dimensional case. The solution Maple obtained was

$$ T = \pmatrix{0 & 1 & -1 & 1\cr -1 & 0 & 1 & 1\cr 1 & -1 & 0 & 1\cr -1 & -1 & -1 & 0\cr}, \ s = 2 $$

This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e_i$ and $v_j$ are the vertices of the unit balls in $\|\cdot\|_1$ and $\|\cdot\|_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e_i\|_\infty \le 1$, $\|T^{-1} v_j\|_1 \le s$ (or equivalently, $\|\text{Adj}(T) v_j\|_1 \le s |\det(T)|$).
Maple's Global Optimization Toolbox returns (almost immediately) an approximate solution, which is (up to roundoff error)

$$ T = \pmatrix{1/3 & 1 & 1\cr -1 & 1 & -1/3\cr 1 & 1/3 & -1\cr},\ s = \dfrac{9}{5}$$

It is easy to check that this does satisfy the constraints. Thus it appears that the answer is $9/5$. It should be possible to prove optimality rigoourously, if somewhat tediously, using the Karush-Kuhn-Tucker conditions.

This can be done as a nonlinear optimization problem: $T$ is a $3 \times 3$ matrix, and if $e_i$ and $v_j$ are the vertices of the unit balls in $\|\cdot\|_1$ and $\|\cdot\|_\infty$ norms respectively, you want to minimize $s$ subject to constraints $\|T e_i\|_\infty \le 1$, $\|T^{-1} v_j\|_1 \le s$ (or equivalently, $\|\text{Adj}(T) v_j\|_1 \le s |\det(T)|$, with $\det(T) \ne 0$).
Maple's Global Optimization Toolbox returns (almost immediately) an approximate solution, which is (up to roundoff error)

$$ T = \pmatrix{1/3 & 1 & 1\cr -1 & 1 & -1/3\cr 1 & 1/3 & -1\cr},\ s = \dfrac{9}{5}$$

It is easy to check that this does satisfy the constraints. Thus it appears that the answer is $9/5$. It should be possible to prove optimality rigoourously, if somewhat tediously, using the Karush-Kuhn-Tucker conditions.