Let $A=(a_{ij})$ be an arbitrary $n \times n$ real symmetric matrix. Let $\| \cdot \|$ denote the operator $2$-norm or equivalently the maximum absolute value of eigenvalues for symmetric matrices. Define

$$d(A) := \min_{D \; \text{is diagonal}} \|A-D\|,$$

as the minimum distance of $A$ from diagonal matrices. What is the best constant $C$  that we can put in the following inequality?

$$d(A) \leq C \max_{i \neq j} |a_{ij}|.$$ 

The constant $C=n-1$ obviously works, by considering the distance of $A$ from its own diagonal, but it can be shown that it's not the smallest constant.