Let notation be as above.

1. (1)

   The function $\vartheta_{\Delta,\ell,c}$ is concave.

2. (2)

   If the family $\{u_{i}\}_{i}$ generates $N_{\mathbb{R}}$, then $\vartheta_{\Delta,\ell,c}$ is strictly concave.

3. (3)

   If $\Delta=\bigcap_{i}\{x\in M_{\mathbb{R}}|\ell_{i}(x)\geq 0\}$, then the restriction of $\vartheta_{\Delta,\ell,c}$ to $\Delta^{\circ}$ is of Legendre type (Definition 3.51).