In order to correctly solve an exercise where I have to calculate the conjugate of the following function $f$, I have to be able to show that it is a convex function.
$$f : \{(x,t) : \| x \| < t\} \subset\mathbb{R}^n \times \mathbb{R} \to \mathbb{R}, \qquad (x,t) \mapsto-\log(t^2 - x'x)$$
I tried to show that the Hessian is positive in the sense of symmetrical matrices (unsuccessful). I also tried the following decomposition:
$$f(x,t) = -\log(\vert t \vert - \| x \|) - \log(\vert t \vert + \| x \|))$$
But I don't see how to conclude from this decomposition. I can't apply a composition rule that would preserve convexity, nor can I make the perspective function of a convex function appear.
Any hint?