Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $f\in C^1(\mathbb R^n\to \mathbb R)$ be a convex function. Suppose the equation

$$f(x+\Delta x)-f(x)-(f'(x),\Delta x) \leq A|\Delta x|^2$$

holds for some constant $A>0$, any $x\in \mathbb R^n$ and any sufficiently small $\Delta x\in \mathbb R^n$. Is it true that the derivative $f'(x)$ must be a (locally) lipschitz function?

Only case $n>1$ is interesting for me (case $n=1$ can be easily verified).

share|improve this question
    
Cross-posted to MSE: math.stackexchange.com/questions/116860/… –  Yemon Choi Mar 6 '12 at 6:51
    
In case $f$ is $C^2$, then the restriction to convexity can be dropped, and in fact, you can show an iff. –  Suvrit Mar 6 '12 at 7:09
3  
Please don't crosspost between here and MSE, without an explanation. –  Scott Morrison Mar 6 '12 at 7:25
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.