Distinguishing between $N$ quantum states
As you mention, it is possible to determine the optimal average success probability numerically, which can be done efficiently via semidefinite programming (see e.g. this paper by Eldar, Megretski and Verghese or these lecture notes by John Watrous), but no closed form expression is known.

However, there are several known upper and lower bounds on the probability of error (i.e. 1 minus the average success probability). In terms of pairwise fidelities, the probability of error in your setting is known to be lower bounded by $\frac{1}{N^2} \sum_{i&gt;j} F(\rho_i,\rho_j)$, and upper bounded by $\frac{2}{N} \sum_{i&gt;j} F(\rho_i,\rho_j)^{1/2}$.

There is also another lower bound known in terms of trace distance: $\frac{1}{2}(1 - \frac{1}{N(N-1)} \sum_{i&gt;j} tr|\rho_i-\rho_j|)$, which reduces to the exact Helstrom bound in the case $N=2$. See this paper for a comparison of all of these, and some other bounds too. Note that all of these bounds hold in the average-case setting where there is a prior probability distribution on the states.