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Let $F:\mathbb{R} \to \mathbb{R}$ be a continuously differentiable function.

It is known that if $F$ has a strict local maximum, then it is not a convex function.

I just would like to ask you for a proper classic reference for that, I mean, a book where I can find that statement proved (or something similar).

ADDITION: I am just asking for a book where all those basic concepts involving convexity are dealt, no matter whether this very specific statement is proved or not.

EDITED: I replaced an absolute maximum with strict local maximum.

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    $\begingroup$ I expect you will find many books proving that the graph of a convex function lies above any of its tangent planes; this result is a trivial consequence of that. And so I very much doubt that you will find a reference writing up this special case as a separate theorem – it is just too easy. $\endgroup$ – Harald Hanche-Olsen Jul 5 '14 at 9:39
  • $\begingroup$ @Harald Hanche-Olsen, that is why I have edited my question to deal with strict local maximum. $\endgroup$ – Vicent Jul 5 '14 at 9:43
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Suppose that $F$ has a strict local maximum at $x^*\in\mathbb R$. Then, there exists some $\varepsilon>0$ such that $f(x^*)>f(x)$ for all $x\in U$, where $U\equiv(x^*-\varepsilon,x^*+\varepsilon)\setminus\{x^*\}$.

To obtain a contradiction, suppose that $F$ is convex. Let $x_1\equiv x^*-\varepsilon/2$ and $x_2\equiv x^*+\varepsilon/2$. Clearly, $x_1,x_2\in U$ and $(x_1+x_2)/2=x^*$. By convexity and strict maximality of $x^*$, $$f(x^*)=f\left(\frac{x_1+x_2}{2}\right)\leq\frac{f(x_1)+f(x_2)}{2}\underbrace{<}_{!!!}\frac{f(x^*)+f(x^*)}{2}=f(x^*),$$ which is a contradiction.

I don't know of any standalone references for this; I just made up the proof myself, using the definitions.

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  • $\begingroup$ Thank you for the answer. It is not exactly what I was asking for, but it actually helps. $\endgroup$ – Vicent Jul 5 '14 at 9:55
  • $\begingroup$ @Vicent Are you trying to locate a particular textbook you remember seeing this result in but can't remember the title of? $\endgroup$ – triple_sec Jul 5 '14 at 9:57
  • $\begingroup$ No, just asking for a good book that deals with this topic. $\endgroup$ – Vicent Jul 5 '14 at 9:59
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    $\begingroup$ Haha, no, it's not, I just inserted it to emphasize the strictness of the inequality, ultimately leading to the contradiction. As for your question, am I right in conjecturing that what you really are looking for is a solid and comprehensive calculus/real analysis textbook? If so, two very popular and great references are Tao and Rudin, but there are many, many others. $\endgroup$ – triple_sec Jul 5 '14 at 10:20
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    $\begingroup$ Thank you! I am going to mark your answer as accepted answer, as you gave the proof and also a reference book in the comments. $\endgroup$ – Vicent Jul 8 '14 at 7:54

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