Positive definite matrix approximation with condition number constraint

Positive definite matrix approximation with a condition number constraint is an optimization problem to find the nearest positive definite matrix whose condition number is smaller than a given constant. We demonstrate that this problem can be converted to a simpler one in this note when we use a unitary similarity invariant norm as a metric. We can especially convert it to a univariate piecewise convex optimization problem when we use a Ky Fan p-k norm. We also present an analytical solution to the problem whose metric is the spectral norm and the trace norm.