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 when we use a unitary similarity invariant norm as a metric, and to a univariate piecewise convex optimization problem when we use a Ky Fan $p$-$k$ norm. We show that we can solve the resulting problem easily by the binary search. We also present an analytical solution to the problem whose metric is the spectral norm and the trace norm.