When to Use Modulus of Convexity and Modulus of Smoothness in Calculations?

[moh salem]

See the pdf file

 

[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
$\rho _{X}(t) =\sup \{\frac{1}{2}(\left\Vert x+y\right\Vert +\left\Vert x-y\right\Vert )-1:\left\Vert x\right\Vert =1,\left\Vert y\right\Vert =t\}$