difference between uniformly convex norms and strictly subadditive norms?

Question

What is the difference between uniformly convex norms and strictly subadditive norms? why we need to define two above concept? how they help us to study Banach spaces? Is the norm induced by inner product simultaneously uniformly convex and strictly subadditive norm?

Answer 1

For a uniformly convex norm the condition of strict subadditivity has to hold uniformly in the following sense;

For any pair of unit vectors $x,y$, the norm of $(x+y)/2$ is strictly less than 1 by an amount that depends only on $|x-y|$. More explicitly, there is an increasing function $\epsilon(r)$ defined for positive $r$, $$\epsilon(r)>0, \quad \lim_{r\to0} \epsilon(r)=0,$$ such that for all $x,y$ in the unit ball $|x|\leq 1, |y|\leq1$, the inequality $$\left|\frac{x-y}{2}\right|\leq 1- \epsilon(|x-y|),$$ holds.

Based on this it should be clear what properties a norm induced by a inner product has.