Suppose $F:\mathbb{R}^n\rightarrow\mathbb{R}$ is a continuous function. Suppose that $F$ attains a local minium in a point $a$. Is true that there exists some ball centered in $a$ such that $F$ restricted to this ball is convex?
Is this Function Convex in a neighbourhood of $a$?
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0Isn't this the definition of a local minimum? – 2012-11-13
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0No, the definition is: $F(a)\leq F(x)$ for all $x$ in a neighbourhood of $a$. – 2012-11-13
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0As long as $a$ is in the interior of your domain, yes. If you draw the tangent line $y=f(a)$, there is a neighborhood such that $f(x)\geq f(a)$; in other words, the function lies above this tangent line. This isn't really a proof, but it's the idea. – 2012-11-13
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0@icurays1 Your intuition is wrong. – 2012-11-16
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0Interesting, I guess this wasn't as obvious as it seemed in my head. Thanks for pointing it out! – 2012-11-16
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