Exercise 2.2

Question

Let $f:\mathfrak{R}^n \mapsto \mathfrak{R}$ be a convex function and let $c$ be some constraint. Show that the set $S = \{\mathbf{x}\in\mathfrak{R}^n\mid f(\mathbf{x}) \leq c\}$ is convex.

Elu Thingol · Accepted Answer

\begin{proof}
Pick an arbitrary $\lambda \in [0,1]$ and $\mathbf{x},\mathbf{y} \in S$. Our aim is to prove that $\lambda \mathbf{x} + (1-\lambda) \mathbf{y} \in S$, or equivalently $f\left(\lambda \mathbf{x} + (1-\lambda) \mathbf{y}ight) \leq c$. By Definition 1.1, we have
\begin{align*}
f\left(\lambda \mathbf{x} + (1-\lambda) \mathbf{y}ight) &\leq
\lambda f(\mathbf{x}) + (1-\lambda) f(\mathbf{y}) &	ext{since $f$ is convex}\
&\leq \lambda c + (1-\lambda)c &	ext{since $\mathbf{x},\mathbf{y} \in S$}\
&= c
\end{align*}
as desired.
\end{proof}

Exercise 2.2

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Exercise 2.2

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