Minimize matrix norm over the unitary matrices

Suppose $$C_1$$ and $$C_2$$ are some fixed $$n \times n$$ matrices. Define the norm $$\| M \| = \sum_{i = 1}^n \max_j |M_{ij}|$$. What is $$\min_U \|C_1 U C_2 \|$$? Here $$U$$ ranges over the $$n \times n$$ unitary matrices. I would also be interested to know what is the minimizer when the norm is one of the "standard" norms, such as the Frobenius norm or the nuclear norm.

I can provide the answer in the case when the norm is the Frobenius norm $$\|\cdot\|_{\text{F}}$$. In this case, $$\|C_1UC_2\|_{\text{F}} = \sqrt{\mathrm{tr}\big((C_1UC_2)(C_1UC_2)^*\big)} = \sqrt{\mathrm{tr}(UC_2C_2^*U^*C_1^*C_1)}.$$ It is a standard result of matrix theory that minimizing this quantity over unitary matrices $$U$$ gives $$\mathrm{min}_U\|C_1UC_2\|_{\text{F}} = \sqrt{\sum_{j=1}^n\lambda_j\mu_{n-j+1}},$$ where $$\lambda_1 \geq \lambda_2 \geq \cdots \geq \lambda_n$$ are the eigenvalues of $$C_1^*C_1$$ and $$\mu_1 \geq \mu_2 \geq \cdots \geq \mu_n$$ are the eigenvalues of $$C_2C_2^*$$ (this follows immediately from Problem III.6.14 in Bhatia's Matrix Analysis book, for example). Furthermore, a unitary $$U$$ that achieves this minimum is one that makes $$UC_2C_2^*U^*$$ diagonal in the same basis as $$C_1^*C_1$$.