(correction) According to your definition of lpdf, you need a function \alpha(||x||) such that V(x)>=\alpha(||x||), \forall x \in B_\epsilon. Do I need to find an \alpha(.)? If not, what's the alternate definition of pdf?

Shreesh P. Mysore, 21 Oct 02

The nature of this definition requires that there exist a function \alpha with the required properties such that V(x)>=\alpha(||x||). This means that to prove that a function V(x) is lpdf or pdf, one needs to find an \alpha that will work for this V(x) (note that the inequality relating V and \alpha is not strict and permits equality) - a simple starting choice for \alpha would be k*||x||², where k is a constant >0.
This is, "unfortunately", the only useful/general definition.