Is it possible to solve the following problem for general norm
$$ \max_{x \geq 0, \|x\| \leq \lambda} y^{\top} x $$
In the special case of infinity norm we have
$$ \begin{aligned} \max_{x \geq 0,\; \|x\|_{\infty} \leq \lambda} y^{\top} x = \max_{x \geq 0,\; x \leq \lambda \boldsymbol{1}} y^{\top} x = \lambda \min_{u \geq 0} \Big\{\boldsymbol{1}^{\top}u: y \leq u\Big\} = \lambda \sum_{\forall i}\max{\{y_i, 0\}} \leq \lambda\|y\|_1 \end{aligned} $$
where $\boldsymbol{1}$ is a vector or all ones. The first equality follows from non-negativity of $x$, the second follows from strong duality of linear program. The last inequality binds only in the special case when $y$ is non-negative.