Let $A=(a_{ij})$ be an arbitrary $n\times n$ real symmetric matrix. Let $||.||$ denote the operator 2-norm or equivalently the maximum absolute value of eigenvalues for symmetric matrices. Define
$$d(A) = \min_{D \; \mathrm{diagonal}} ||A-D||,$$
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 it's own diagonal. But it can be shown that it's not the smallest constant.