Fix a natural number $n\geq 1.$ Let $\mu$ be a norm on $\mathbb{R}^n$ satisfying $$\mu(0,...,0,\stackrel{i}{1},0,...,0) = 1 \quad\text{for all }1\leq i\leq n.$$ Let $$B_{\mu} = \{(a_1,...,a_n)\in \mathbb{R}^n : \mu(a_1,...,a_n)\leq 1\},$$ that is, ball of $\rho$-norm centered at origin. Let $$B_{\ell^1} = \{(a_1,...,a_n)\in\mathbb{R}^n: |a_1|+\cdots|a_n|\leq 1\}.$$

One can show easily that $B_{\ell^1}\subseteq B_{\mu}$ by using triangle inequality and above equation.

Define dual norm $\rho^*$ on $\mathbb{R}^n$ by $$\mu^*(a_1^*,...,a_n^*) = \sup_{(a_1,...,a_n)\in B_{\mu} }\bigg| \sum_{i=1}^n a_i^*a_i \bigg|.$$

Question: Assuming that $\mu \neq \|\cdot\|_\infty$-norm. Fix $(a_1,...,a_n)\in \mathbb{R}^n$ with $\mu(a_1,...,a_n)=1.$ Is it always true that there exists an extreme point $(a_1^*,...,a_n^*)$ of $B_{\mu^*}$ such that $a_i^*\neq 0$ for all $1\leq i\leq n$ and $$\sum_{i=1}^n a_i^*a_i=1?$$

Recall the following well-known result:

If $X$ is a Banach space, then for any $x\in X,$ there exists an extreme point $x^*$ of $B_{X^*}$ such that $x^*(x) = \|x\|.$

However, if we let $X = \mathbb{R}^n,$ my question requests one more condition on $x^*,$ that is, all its component is nonzero. Can this be done?