For a convex function $f$, is the following set convex: $X := \{x ∶ -f(x) \leq 1 \}$?
I know that the set $X := \{x ∶ f(x) \leq 1\}$ is convex, but I'm unsure about the $-f(x)$ in the first set.
2026-03-26 13:30:14.1774531814
For a convex function $f$, is the following set convex: $X := \{x ∶ -f(x) \leq 1 \}$?
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Not necessarily. Consider $f(x) = x^2 - 2$. This set would be $\{x : 2 - x^2 \leq 1\}$, i.e. $\{x : x^2 \geq 1 \}.$ But this set is not convex; for example, $-1$ and $1$ are both in the set, yet $\frac{1}{2} (-1) + \frac{1}{2} (1) = 0$ is not.