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Let $C$ be a nonempty subset of a real Hilbert space $H$. Let $\lbrace z_n \rbrace_{n=1}^{\infty} \subset H$ be Fejer monotone with respect to $C$. Show that $\lbrace z_n \rbrace_{n=1}^{\infty} $ is also Fejer monotone with respect to the convex hull of $C$.

Let $\lbrace z_n \rbrace_{n=1}^{\infty} \subset H$ be Fejer monotone with respect to $C$. Then $\forall x \in C$ and $\forall n \in \mathbb{N}$ we have $$ \|z_{n+1} - x \| \leq \|z_{n} - x \|.$$

The convex hull of $C$, denoted by $conv(C)$, is the set $$conv(C)= \big\lbrace \sum_{k=1}^{m} \alpha_k x_k : x_1,x_2,...,x_m \in C, \alpha_1, ..., \alpha_m >0, \sum_{k=1}^{m} \alpha_k =1, m \in \mathbb{}N \big\rbrace .$$

I need to show that for every $ \sum_{k=1}^{m} \alpha_k x_k \in conv(C)$ and for every $n \in \mathbb{N}$, $$ \|z_{n+1} - \sum_{k=1}^{m} \alpha_k x_k \| \leq \|z_{n} - \sum_{k=1}^{m} \alpha_k x_k \|$$ $$\iff \|z_{n+1} - \sum_{k=1}^{m} \alpha_k x_k \|^2 \leq \|z_{n} - \sum_{k=1}^{m} \alpha_k x_k \|^2 .$$

Please help.

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It holds $$ \|z_{n+1}-x\| \le \|z_n-x\| $$ if and only if $$ 0\le \|z_n-x\|^2 - \|z_{n+1}-x\|^2 = \|z_n\|^2 - \|z_{n+1}\|^2 - 2(x,z_n-z_{n+1}). $$ This latter inequality is affine linear in $x$. Hence if it holds for $x\in C$ then it holds for all $x\in conv(C)$.

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