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

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SUMMARY

The discussion focuses on the properties of norms that satisfy the reverse triangle inequality, specifically the condition ||x+y|| ≥ ||x|| + ||y||. The initial suggestion of p-norms for 0 < p < 1 is corrected, as these do not meet the criteria. The conversation shifts towards exploring the concept of a "norm" with a unit ball that is concave rather than convex, emphasizing the impossibility of having a unit ball that lacks a line segment between two points due to central symmetry.

PREREQUISITES
  • Understanding of normed vector spaces
  • Familiarity with p-norms and their properties
  • Knowledge of convex and concave functions
  • Basic concepts of central symmetry in geometry
NEXT STEPS
  • Research the properties of norms in non-convex spaces
  • Explore the implications of the reverse triangle inequality in functional analysis
  • Study the characteristics of concave functions and their applications
  • Investigate the concept of unit balls in various normed spaces
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Mathematicians, students of functional analysis, and researchers interested in the properties of norms and their geometric implications.

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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?
 
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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.
 
Last edited:
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)
 

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