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Weird characterization of Lipschitz functions through convex functions

Lipschitz in $mathbb R^1$ implies Lipschitz along any line in $mathbb R^k$ (for convex functions)$F(x) = f(x) + g(x) + h(x)$, where h(x) is strongly convex , is also strongly convexCharacterization convex function.Strictly convex setAny example of strongly convex functions whose gradients are Lipschitz continuous in $mathbbR^N$Lipschitz constant of the convex function $f(x) - fraca2 |x|^2$Convex and continuous function on compact set implies Lipschitzsufficient conditions for differentiability of convex conjugateStatement about Lipschitz functionsWhat is the condition to have a Lipschitz continuous the gradient for convex function?

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$begingroup$

I'm stuck with this problem:

Let $L: mathbbR longrightarrow mathbbR$ be a strictly convex function and assume that the function $u : mathbbR longrightarrow mathbbR$ satisfies $$L(u(x)) + L(u(y)) leq L left (fracu(x)+u(y)2 + fracx-y2 right ) + L left (fracu(x)+u(y)2 - fracx-y2 right )$$
for all $x,y in mathbbR$. Prove that the function $u$ is Lipschitz continuous.

Hint: Use strict convexity about the point $(u(x)+u(y))/2$

Even using the hint I can't find any way of approaching the problem so any help would be greatly appreciated. Thanks!

real-analysis convex-analysis lipschitz-functions

share|cite|improve this question

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244

$endgroup$

add a comment | 

0

$begingroup$

I'm stuck with this problem:

Let $L: mathbbR longrightarrow mathbbR$ be a strictly convex function and assume that the function $u : mathbbR longrightarrow mathbbR$ satisfies $$L(u(x)) + L(u(y)) leq L left (fracu(x)+u(y)2 + fracx-y2 right ) + L left (fracu(x)+u(y)2 - fracx-y2 right )$$
for all $x,y in mathbbR$. Prove that the function $u$ is Lipschitz continuous.

Hint: Use strict convexity about the point $(u(x)+u(y))/2$

Even using the hint I can't find any way of approaching the problem so any help would be greatly appreciated. Thanks!

real-analysis convex-analysis lipschitz-functions

share|cite|improve this question

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244

$endgroup$

add a comment | 

$begingroup$

I'm stuck with this problem:

Let $L: mathbbR longrightarrow mathbbR$ be a strictly convex function and assume that the function $u : mathbbR longrightarrow mathbbR$ satisfies $$L(u(x)) + L(u(y)) leq L left (fracu(x)+u(y)2 + fracx-y2 right ) + L left (fracu(x)+u(y)2 - fracx-y2 right )$$
for all $x,y in mathbbR$. Prove that the function $u$ is Lipschitz continuous.

Hint: Use strict convexity about the point $(u(x)+u(y))/2$

Even using the hint I can't find any way of approaching the problem so any help would be greatly appreciated. Thanks!

real-analysis convex-analysis lipschitz-functions

share|cite|improve this question

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244

$endgroup$

share|cite|improve this question

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244

share|cite|improve this question

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244

asked Mar 16 at 10:29

Lorenzo LiveraniLorenzo Liverani

244