For $q$-dimensional data, penalized versions of the sample covariance matrix are important when the sample size is small or modest relative to $q$. Since the negative log-likelihood under multivariate normal sampling is convex in $Σ^{-1}$, the inverse of its covariance matrix, it is common to add to it a penalty which is also convex in $Σ^{-1}$. More recently, Deng-Tsui (2013) and Yu et al.(2017) have proposed penalties which are functions of the eigenvalues of $Σ$, and are convex in $\log Σ$, but not in $Σ^{-1}$. The resulting penalized optimization problem is not convex in either $\log Σ$ or $Σ^{-1}$. In this paper, we note that this optimization problem is geodesically convex in $Σ$, which allows us to establish the existence and uniqueness of the corresponding penalized covariance matrices. More generally, we show the equivalence of convexity in $\log Σ$ and geodesic convexity for penalties on $Σ$ which are strictly functions of their eigenvalues. In addition, when using such penalties, we show that the resulting optimization problem reduces to to a $q$-dimensional convex optimization problem on the eigenvalues of $Σ$, which can then be readily solved via Newton-Raphson. Finally, we argue that it is better to apply these penalties to the shape matrix $Σ/(\det Σ)^{1/q}$ rather than to $Σ$ itself. A simulation study and an example illustrate the advantages of applying the penalty to the shape matrix.