Let $x^{(1)}, \ldots, x^{(N)}$ be independent samples from an $\mathcal N(\mu,\Sigma)$ distribution, where it is known that $\Sigma^\mathrm{min} \preceq \Sigma \preceq \Sigma^\mathrm{max}$, where $\Sigma^\mathrm{min},~ \Sigma^\mathrm{max}$ are given positive definite matrices.
The associated log-likelihood function $\ell(\mu,\Sigma)$ (including the constraint on $\Sigma$) is concave.