This is a follow up question to this:
Norm of diference of matrices of different rank
Suppose $A$ is a norm one $n\times n$ matrix of rank $k$. What is $\inf\|A-B\|$, where the infimum is taken over all norm one $n\times n$ matrices of rank $p<k$. Is there some closed formula that depends on $n, k, p$ and the eigenvalues of $A$?
Without the requirement $\|B\|=1$ the exact value of the infimum is known: $$ \inf \{\|A-B\| \colon \operatorname{rank} B = p\} = \sigma_{p+1} $$ where $\sigma_1\ge \dots \ge \sigma_n$ are the singular values of $A$, in nonincreasing order. This is the min-max principle for singular values.
If you require $\|B\|=1$, the singular value $\sigma_{p+1}$ still gives a lower bound on $\|A-B\|$, but it need not be the infimum.