My Math Forum convex combination of closed and convex sets

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 April 6th, 2009, 11:31 AM #1 Newbie   Joined: Apr 2009 Posts: 1 Thanks: 0 convex combination of closed and convex sets Given: $V$ is a vector space over the real numbers and $P_1, \ldots, P_N$ are convex and closed sets in V. $U$ is a closed and convex subset of $\{ u=(u_1,\ldots,u_N) \in \mathbb{R}^N \: \mid \: 0 \leq u_n \leq 1, \: \sum_{n=1}^N u_n = 1 \}$. My question : how to proof that $P:=\bigcup_{u \in U} \sum_{n=1} u_n P_n$ is a closed convex set? You can write $P$ as the image of the following bilinear function: $f:\: U \times \prod_{n=1}^N P_n \rightarrow V$ defined by $f(u,p)= \sum_{n=1}^N u_n p_n$, maybe it follows from properties of this function? Thanks in advance for any help.

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