Convexity of sublevel set $\sin (x) \leq 1$

Question

I am having trouble determining if the following set is convex

\begin{equation} \left\{x \in \mathbb{R} : \sin (x) \leq 1 \right\} \end{equation}

I know that the function itself is not convex function but on the other hand, a $\sin x$ is less than or equal $1$ on all periods so it creates a continuous line as a set which is a convex set. What is the convexity of this constraint and what is the proof ?

none right? $[0, 2\pi]$ will always be $\leq$ 1, so it is convex? — Praditya Raudi, Mar 08 '21 at 01:06
I see, so sin(x) is non convex function but it still can be a constraint for convex set. Thank you! — Praditya Raudi, Mar 08 '21 at 01:11

score 1 · Accepted Answer · answered Mar 08 '21 at 01:09

1

The set $A = \{x: x \in \mathbb{R}, \sin x \le 1\}= (-\infty,\infty)$ which is clearly convex.

answered Mar 08 '21 at 01:09

Convexity of sublevel set $\sin (x) \leq 1$

1 Answers1