What are the equality conditions for proving strict convexity?

[member 428835]

Hi PF!

Do you know what a strictly convex function is? I understand this notion in the concept of norms, where in the plane I've sketched the $L_1,L_2,L_\infty$ norms, where clearly $L_1,L_\infty$ are not strictly convex and $L_2$ is. Intuitively it would make sense that any $L_1,L_\infty$ function is not strictly convex (similar to it's norm) and $L_2$ functions are, but how would you even show this?

 

[StoneTemplePython]

Lp norms have 3 things
(1.) symmetry
(2.) homogeneity with respect to scaling by positive numbers -- consider in particular $p \in (0,1)$
(3.) triangle inequality (or sub additivity)

if you carefully apply (2.) and (3.) you recover a definition of convexity. As far as strictness of the convexity, what do you know about the proof behind (3.), and in particular the equality conditions underlying it? The typical way is via Hoelder, but there are clever other ways. A more pedestrian approach for this particular problem is to come up with counterexamples on strictness for $L_1$ and $L_\infty$ and re-examine the equality conditions of Cauchy-Schwarz.