Convex support of an exponential family ($C=\overline{\mathrm{conv}(\mathrm{supp}\nu )}$)

The convex support is the closed convex hull of the canonical sufficient-statistic support. For a minimal exponential family it has nonempty interior. Means at finite interior natural parameters lie in that interior, because a supporting hyperplane at a boundary mean would force the sufficient statistic itself onto that hyperplane.