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Proving a set is convex

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Feb 15, 2012
Let $P=co(x_{1},....,x_{n})$ be a convex hull in a normed space X. Define the set
M={x in x | d(x,P)< s } for some s>0. I want to show M is convex. I was wondering whether there any sufficent conditions relating to compactness or closedness for a set to be convex. Otherwise I need to show directly, which I have tried to do:

let x,y be in M and 0<t<1.

Then d(tx+(1-t)y,P)=inf{d(tx+(1-t)y,p)| p in P}
<_ inf{||tx-p||+ ||(1-t)y-p||