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I am reading the book Methods of Modern Mathematical Physics by Reed & Simon. In one of their lemma they claim that for a locally convex space $X$,

Let $V$ be an open, convex, balanced subsets of $X_1$, where $X_1$ is a subspace of $X$ inheriting the subspace topology. Let $O$ be an open set such that $O\cap X_1 =V$, $O_1$ an open, convex, balanced subsets of $O$. Then $$ Z:=\{\alpha x+\beta y\ |\ x\in O_1, y\in V, |\alpha|+|\beta|=1 \} $$ is an open, convex, and balanced subset of $X$.

I can see that $Z$ is open and that it is balanced, but I failed to see how it is convex. I would really appreciate if anyone could explain this to me (or give some hint if it is actually not that hard).

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    Hint: An alternate description of $Z$ is $$Z = \{\alpha x + \beta y \mid x \in O_1, y \in V, \lvert\alpha\rvert + \lvert\beta\rvert \leqslant 1\}.$$ – 2017-02-03
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    I was also thinking that if $|\alpha|+|\beta|\le 1$ would make this a lot easier. I forgot that convexity plus balance imply that fact. Thank you! – 2017-02-03
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    Another way to see it is to note that since $O_1$ and $V$ are balanced, we have $$Z = \{ t x + (1-t)y \mid x \in O_1, y\in V, t\in [0,1]\}.$$ – 2017-02-03

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