Can we say anything about the minimum of a perspective function compared to that of the original function?

Question

Given convex function $f(x)$, its perspective function is $g(x,t) = tf(x/t), t>0$ is also convex. Is the minimum of $g$ over $(x,t)$ always less than (or larger than) the minimum of $f(x)$? Note that $x \in \mathbb{R}^n, t \in \mathbb{R}^1$.

Answer 1

Yes, the minimum of the perspective is always less than or equal to the minimum of the original function. Suppose $x^*$ is a minimizer of $f$, then $$ \min_x f(x) = f(x^*) = g(x^*,1) \ge \min_{x,t} g(x,t) $$