The motivation for this question comes from this paper where we study lower bounds on the complexity of convex optimization algorithms (BTW, if you find a better title for my question, please let me know).

Some notation: let $X=B_p^n$ be the unit ball of $(\mathbb{R}^n,\|\cdot\|_p)$, and let $Y=B_{q_{\ast}}^n$ be the unit ball of $(\mathbb{R}^n,\|\cdot\|_{q_{\ast}})$, where $2\leq p,q \leq \infty$ and $1/q+1/q_{\ast}=1$ (in the paper we consider the case $p=q$).

Let now $T\leq n$. One of the key ingredients of our analysis depends on finding linear functionals $\xi^t\in Y$, $t=1,\ldots,T$, such that we can lower bound, for arbitrary $s_t=\pm 1$, the quantity $$ \Delta_{p,q}^{T,n} = -\min_{x\in X} \max_{1\leq t\leq T} s_t \langle \xi^t,x\rangle. $$ In the case $p=q$ this is done by simply picking $\xi^t$ to be the first $T$ canonical vectors, which gives $\Delta_{p,p}^{T,n}\geq 1/T^{1/p}$. My question is whether there exist good lower bounds when $p\neq q$.

In general, by simple dualization arguments, this turns out to be equivalent to $$ \Delta_{p,q}^{T,n} = \min_{\lambda \in \partial B_1^T} \left\|\sum_{t=1}^T \lambda_t \xi^t \right\|_{p_{\ast}}, $$ where $\partial B_1^T$ denotes the boundary of the $\|\cdot\|_1$-unit ball. Therefore, we are just looking for a linear transformation $\Xi:B_1^T\to B_{p_{\ast}}^n$ (in matrix form: $\Xi=[\xi^1|\ldots|\xi^T]$, where each $\|\xi^t\|_{q_{\ast}}=1$) with $\mbox{ker}(\Xi)=0$, and with the image of the boundary of the cross-polytope as far as possible from the origin.

This question looks remarkably similar to some computations for $\ell_p$ embeddings in the local theory of Banach spaces. However, this is far from my expertise, so I haven't been able to extract useful ideas from there.

If you see some explicit connection with something that is known (or figure out a simple way to lower bound $\Delta$), please let me know.