Calculate $k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$ for a special $\ell$ function

Question

We consider the function $\ell:\mathbb{R}^{m}\rightarrow \mathbb{R}$ given by $$\ell(\xi):=-\max\left\{-\left\langle x,\xi\right\rangle+10 \tau, -51\left\langle x,\xi\right\rangle -40\tau \right\}$$ where $x\in\mathbb{R}^{m}_{+}$ with $\sum_{i=1}^{m}x_{i}=1$, and $\tau\in\mathbb{R}$ are fixed .

Question: Calculate $$k:=\sup\left\{\left\|\theta\right\|_{*} \: |\: \ell^{*}(\theta)<\infty\right\}$$ where for $\theta\in\mathbb{R}^{m}$, $\left\|\theta\right\|_{*} =\sup_{\xi\in \mathbb{R}^{m}}\left\langle \theta,\xi\right\rangle $ and $$\ell^{*}(\theta)=\sup_{\xi\in \mathbb{R}^{m}}\left[\left\langle \theta,\xi\right\rangle - \ell(\xi)\right] .$$ (we recall $\left\langle \theta,\xi\right\rangle =\theta^{t}\xi$, and in this context we consider $\left\|\cdot\right\|$ as 1-norm)

Remark: In my attempt I got $ k = 0 $, but I do not trust my answer since within the context in which this question is found this answer is not realistic. This is a important step for my thesis, I ask for your help through any suggestion or response.

Note that in this context $\left\|\cdot\right\|_{*}$ is $\infty$-norm.

Answer 1

The function $\ell$ is concave (it's a minimum of linear functions). Its conjugate $\ell^*(\theta)$ is the supremum over a linear function minus $\ell$. If I see correctly, $\ell^*(\theta) = \infty$ for any $\theta$ and thus, $k$ is the supremum of the empty set, usually set to be $-\infty$.