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?
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\} ##