What Are the Properties of Norms Satisfying the Reverse Triangle Inequality?

[Office_Shredder]

I'm interested in thing that are norms except for the fact that they satisfy the reverse triangle inequality $$||x+y|| \geq ||x|| + ||y||$$. The obvious example is taking p-norms for 0<p<1. Does anyone know of others or if there's any theory developed on this topic?

 

[eok20]

Are you requiring the norm to be positive definite? If so, I'm pretty sure that your space can have only one point.EDIT: the p norms don't satisfy the reverse triangle inequality: take x = -y.

 

[Office_Shredder]

Oops you're right. I was reading on wikipedia and misinterpreted something.

I should have just stuck with what I originally wanted, which is a "norm" whose unit ball is as concave as possible, rather than convex. Obviously you can't have a unit ball that never contains a line between two points (since it's centrally symmetric)