Let $1\leq i\leq r$ and consider the affine map $\ell_{i}\colon\Delta\to\mathbb{R}_{\geq 0}$. We have that $-z\log(z)$ is a strictly concave function on $\mathbb{R}_{\geq 0}$ and $-\ell_{i}\log(\ell_{i})=\ell_{i}^{*}(-z\log(z))$. Hence, each function $-c_{i}\ell_{i}(x)\log(\ell_{i}(x))$ is concave and so is $\vartheta$, as stated in (1)

For statement (2), let $x_{1},x_{2}$ be two different points of $\Delta$. The assumption that $\{u_{i}\}_{i}$ generates $N_{\mathbb{R}}$ implies that $\ell_{i_{0}}(x_{1})\neq\ell_{i_{0}}(x_{2})$ for some $i_{0}$. Hence, the affine map $\ell_{i_{0}}$ gives an injection of the segment ${\overline{x_{1}x_{2}}}$ into $\mathbb{R}_{\geq 0}$. We deduce that $-c_{i_{0}}\ell_{i_{0}}\log(\ell_{i_{0}})$ is strictly concave on ${\overline{x_{1}x_{2}}}$ and so is $\vartheta$. Varying $x_{1},x_{2}$, we deduce that $\vartheta$ is strictly concave on $\Delta$.

For statement (3), it is clear that $\vartheta|_{\Delta^{\circ}}$ is differentiable. Moreover, the assumption that $\Delta$ is the intersection of the halfspaces defined by the $\ell_{i}$’s implies that the $u_{i}$’s generate $N_{\mathbb{R}}$ and so $\vartheta$ is strictly concave. The gradient of $\vartheta$ is given, for $x\in\Delta^{\circ}$, by

Let $\|\cdot\|$ be a fixed norm on $M_{\mathbb{R}}$ and $(x_{j})_{j\geq 0}$ a sequence in $\Delta^{\circ}$ converging to a point in the border. Then there exists some $i_{1}$ such $\ell_{i_{1}}(x_{j})\stackrel{{\scriptstyle j}}{{\to}}0$. Thus, $\|\nabla\vartheta(x)\|\stackrel{{\scriptstyle j}}{{\to}}\infty$ and the statement follows. ∎