(1) and (2) are consequences of (3) and (5) in Lemma 3.14, respectively.

(3) We set

$$f_{0}(t_{1},\ldots,t_{r}):=\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}}(x)$$

for $(t_{1},\ldots,t_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$. By (1) and (2), for $\lambda\in[0,1]\cap{\mathbb{Q}}$ and $(t_{1},\ldots,t_{r}),(t^{\prime}_{1},\ldots,t^{\prime}_{r})\in(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$, we have

$$f_{0}(\lambda(t_{1},\ldots,t_{r})+(1-\lambda)(t^{\prime}_{1},\ldots,t^{\prime}_{r}))\\ \hskip-50.00008pt=\mu^{\operatorname{a}}_{(\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}})^{\otimes\lambda}\otimes(\overline{L}\otimes\overline{L}_{1}^{\otimes t^{\prime}_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t^{\prime}_{r}})^{\otimes(1-\lambda)}}(x)\\ \leq\lambda\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t_{r}}}(x)+(1-\lambda)\mu_{\overline{L}\otimes\overline{L}_{1}^{\otimes t^{\prime}_{1}}\otimes\cdots\otimes\overline{L}_{r}^{\otimes t^{\prime}_{r}}}(x)\\ =\lambda f_{0}(t_{1},\ldots,t_{r})+(1-\lambda)f_{0}(t^{\prime}_{1},\ldots,t^{\prime}_{r}),$$

that is, $f_{0}$ is concave on $(I_{1}\times\cdots\times I_{r})\cap{\mathbb{Q}}^{r}$. Therefore, the assertion (3) follows from [7, Corollary 1.3.2]. ∎