# Convexity and smoothness

1. Dec 14, 2014

### moh salem

See the pdf file

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• ###### convexity.pdf
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2. Dec 19, 2014

### Greg Bernhardt

Thanks for the post! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post?

3. Dec 20, 2014

### moh salem

Question: What do you mean the following concepts? and when to use?
Let $\delta _{X}:[0,2]\longrightarrow \lbrack 0,1]$ }be the
modulus of convexity of X, and defined by
$\delta _{X}(\varepsilon )=\inf \{1-\frac{1}{2}\left\Vert x-y\right\Vert :\left\Vert x\right\Vert \leq 1, \left\Vert y\right\Vert \leq 1, \left\Vert x-y\right\Vert \geq \varepsilon \}.$
And let $\rho _{X}:[0,\infty \lbrack \longrightarrow \lbrack 0,\infty \lbrack$
be the modulus of smoothness of X, and defined by
$\rho _{X}(t) =\sup \{\frac{1}{2}(\left\Vert x+y\right\Vert +\left\Vert x-y\right\Vert )-1:\left\Vert x\right\Vert =1,\left\Vert y\right\Vert =t\}$

Last edited by a moderator: Dec 30, 2014