A property of infimum??

Question

Let $X$ be some space (eg. vector space or Banach space).

When is it true that: for any $\epsilon >0$ small, there exists an $f \in X$ such that $$(1+\epsilon) \inf_{g \in X} I(g) \geq I(f)?$$ Here $I:X \to \mathbb{R}$.

Answer 1

This is true whenever $$\inf_{g \in X}I(g) > 0 \ \ \ \mbox{ or } \ \ \ \min_{g \in X}I(g) = 0$$ to see this, simply apply the definition of $\inf$.

On the contrary, if $\inf_{g \in X}I(g) < 0$, you cannot find such an $f$ because otherwise $$I(f) \le (1+ \epsilon) \inf_{g \in X}I(g) < \inf_{g \in X}I(g) \le I(f)$$ a contradiction.