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So all I have is that $\mathscr{H}$ is a Hilbert space and that $f:\mathscr{H}\to\mathbb{R}$ is a convex function.

i.e. for all $x,y\in\mathscr{H}$ and $\alpha\in[0,1]$, $f(\alpha x +(1-\alpha)y)\leq \alpha f(x)+(1-\alpha)f(y)$.

Define $C = \left\{x\in\mathscr{H}:f(x)\leq 1\right\}$. Is $C$ a convex set?

i.e. $\alpha x+(1-\alpha)y \in C $ when $x,y \in C$ and $0\leq\alpha\leq 1$.

user100928
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1 Answers1

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Yes, $C$ is always convex. This can be proved by a direct calculation, using the definition of convexity.

gerw
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