I want to prove that the following composed function $g \circ L$ is always (strictly) convex :
\begin{alignat*}{3}
&g&&(t&&) && =-\log(1-e^{-t}) \qquad && \text{where }t\in\mathbb{R}\\
&L&&(A&&) && = X^TAX \qquad &&\text{where }X \in \mathbb{R}^n \quad \text{and} \quad A \> \text{is } (n \times n) \> \text{matrix}
\end{alignat*}
It would be convenient for me to use the definition of convexity $\textit{i.e.}$ $$g \circ L \>(\lambda A+(1-\lambda)B) \leq \lambda \> g\circ L(A) + (1-\lambda) \> g \circ L(B)$$
Beginning of the proof:
We see that $L$ is a linear function so $$g \circ L \>(\lambda A+(1-\lambda)B) = g \circ (\lambda L(A) +(1-\lambda)L(B))$$
Best Answer
\begin{alignat*}{3} g \circ L \> (\lambda A +(1-\lambda ) B) &= g\circ (\lambda L(A) + (1-\lambda )L(B)) \qquad && \text{by linearity of $L$} \\ & = \lambda \>g\circ L\>(A) + (1-\lambda)\>g\circ L \> (B) \quad &&\text{by convexity of $g$} \end{alignat*}
which proves the convexity of $g\circ L$