= Solution
A <natural exponential family> of order $p$ has an observation vector $Y\in\mathbb R^p$ and <probability density function>, relative to a fixed reference measure $\nu$,
$$
f_\theta(y)=h(y)\exp\{\theta^Ty-\kappa(\theta)\},\qquad \kappa(\theta)=\log\int h(y)e^{\theta^Ty}\,d\nu(y).
$$
Its <natural parameter space> is $\mathcal N=\{\theta\in\mathbb R^p:\kappa(\theta)<\infty\}$. A <full natural exponential family> allows every $\theta\in\mathcal N$. A <regular natural exponential family> has an open parameter space; fullness and regularity together mean that $\mathcal N$ itself is open. A <minimal exponential family> additionally requires that no nonzero <linear combination> of the components of $Y$ be constant <almost surely>. This is the condition that the genuinely needed number of canonical <sufficient statistics> is $p$.
<Hölder's inequality> shows that $\mathcal N$ is convex: for $0<t<1$, its normalizing integral at $t\theta+(1-t)\eta$ is at most the product of the respective integrals to powers $t$ and $1-t$. Taking logarithms also proves <convexity> of the <cumulant function> $\kappa$. In the minimal case its <Hessian matrix> is positive definite on the interior, as the <covariance> calculation below establishes. Keeping fullness, regularity and minimality separate matters when discussing existence or uniqueness of a <maximum-likelihood estimator>.
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