Rosen's unique equilibrium conditions: Multi dimensional strategies?

I was wondering if the uniqueness of equilibrium conditions in n-person games as published in Rosen's 1965 paper (J. B. Rosen. Existence and uniqueness of equilibrium points for concave n-person games. Econometrica, 1965) are for single-dimensional strategy spaces only? It says in the paper that the strategy of each player can be a vector, but the rest of the conditions seem to be derived for one-dimensional strategy spaces. Can the results be generalized to multi-dimensional spaces as well? Thank you.

These results can indeed be extended to a more general setting. In this case, define the $$\nu$$-th player's strategy vector $$x^\nu \in R^{n_\nu}$$ and so $$x=(x^1,\dots,x^N)^T$$ of a size $$n_1+\dots+n_N$$ for a $$N$$ player game.
In this context, $$\nabla_\nu \varphi_\nu(x)$$ is a vector of size $$n_\nu$$.
The pseudo-gradient (denoted $$g(x,r)$$ in the paper) is still the vector concatenated by all the gradients of $$\varphi$$ i.e. $$g(x,r)=(\nabla_1 \varphi_1(x),\dots,\nabla_N \varphi_N(x))^T$$. In this general case, the positivity constraint in the definition of diagonally strictly concave means that each component of a vector is positive.