It is known that (given a $\sigma$-finite Borel reference measure $\nu$ on $\mathbb{R}$) the parameter space of an exponential family is convex in Euclidean space. However, my question is, for an the associated exponential manifold $\sqrt{EM(c)}$ a convex subset of $L^2_{\nu}$? Where:

- $$ EM(c)\triangleq \left\{ p(\cdot,\theta):\, \theta \in \Theta,\, p(x,\theta)=exp(\theta^Tc(x)-\psi(\theta)) \right\}, $$ for some fixed continuously-differentiable scalar functions $c=\{c_1,\dots,c_n\}$ on $\mathbb{R}^d$,
- We assume that $EM(c)\subseteq L^2_{\nu}$.