# Show that the function is convex

Show that the function $$f: S \to \mathbb R$$ given by $$f(x,s,t):=-\ln(st - ||x||^2)$$ is convex on $$S := \left\{(x,s,t) \in \mathbb R^n \times \mathbb R \times \mathbb R: \frac{\|x\|^2}{s}0, t>0 \right\}$$

$$-\ln(x)$$ is a convex monotonic decreasing function, hence we can use composition by showing the inner part is concave. Also we know $$-||x||^2$$ is concave and $$st$$ is neither convex nor concave on $$S$$. Also in $$S$$ we have $$st>||x||^2$$ but I was unable to show the convexity of the function as a whole

• What are your thoughts on this? What have you tried already (perhaps, what do you know about the convexity of $x^2$ and $-\ln x$?) Questions without visible effort or context tend to get downvoted. – postmortes Feb 15 at 10:43

$$-\ln(st-\| x\|^2)=-\ln(s(t-\frac{| x\|}{s}^2))=-\ln(s)-\ln(t-\frac{| x\|}{s}^2)$$