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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?

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Isn't this the definition of a local minimum? –  espen180 Nov 13 '12 at 17:17
    
No, the definition is: $F(a)\leq F(x)$ for all $x$ in a neighbourhood of $a$. –  Tomás Nov 13 '12 at 17:19
    
As 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. –  icurays1 Nov 13 '12 at 17:25
    
@icurays1 Your intuition is wrong. –  Did Nov 16 '12 at 21:49
    
Interesting, I guess this wasn't as obvious as it seemed in my head. Thanks for pointing it out! –  icurays1 Nov 16 '12 at 22:03
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1 Answer

up vote 2 down vote accepted

Investigate the behavior of $x^2 (\sin^2\frac1x + 1)$ at $0$.

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Thank you azrael. –  Tomás Nov 13 '12 at 19:01
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