Extension of the Triangle Inequality

AI Thread Summary
The discussion centers on the extension of the triangle inequality, specifically the assertion that |a + b|^p ≤ 2^p (|a|^p + |b|^p) for p > 1. Participants express uncertainty about the validity of this inequality and share attempts at proving it, including using induction and the binomial theorem. One contributor notes that the inequality can be derived from the convexity of the function x^p for p > 1. The conversation highlights the complexity of proving the inequality and seeks confirmation of its truth. The conclusion remains that further exploration is needed to definitively establish the inequality's validity.
Caspian
Messages
15
Reaction score
0
Pretty much knows the triangle inequality.
\left| a + b \right| \le \left| a \right| + \left| b \right|

I was reading a source which asserted the following extension of the triangle inequality:
\left| a + b \right|^p \le 2^p \left(\left| a \right|^p + \left| b \right|^p\right)

This is bugging me because I can't figure out how to prove or disprove it. It's sensible enough that it might actually be true... but want to know for sure whether it's true.

I tried proving it by induction, but it got really messy...

Is this inequality true? Or, is it wrong? Thanks!
 
Mathematics news on Phys.org
when you tried induction, did you try the binomial theorem?
 
It follows from convexity of xp for p > 1.
<br /> (\frac{1}{2}|a|+\frac{1}{2}|b|)^p\leq \frac{1}{2}|a|^p+\frac{1}{2}|b|^p<br />
<br /> (|a|+|b|)^p\leq 2^{p-1}(|a|^p+|b|^p)<br />
 
Suppose ,instead of the usual x,y coordinate system with an I basis vector along the x -axis and a corresponding j basis vector along the y-axis we instead have a different pair of basis vectors ,call them e and f along their respective axes. I have seen that this is an important subject in maths My question is what physical applications does such a model apply to? I am asking here because I have devoted quite a lot of time in the past to understanding convectors and the dual...
Insights auto threads is broken atm, so I'm manually creating these for new Insight articles. In Dirac’s Principles of Quantum Mechanics published in 1930 he introduced a “convenient notation” he referred to as a “delta function” which he treated as a continuum analog to the discrete Kronecker delta. The Kronecker delta is simply the indexed components of the identity operator in matrix algebra Source: https://www.physicsforums.com/insights/what-exactly-is-diracs-delta-function/ by...
Back
Top