Here is what I mean exactly: 


Let $A=(a\_1,a\_2)$ and $B=(b\_1,b\_2)$ be two points in the real plane (for simplicity, but general finite dimensions would also be nice), and define the $\ell\_p$-metric as $\|A\|\_p=(|a\_1|^p+|a\_2|^p)^{1/p}$ for $1\leq p<\infty$ and set $\|A\|\_\infty=\max(a\_1,a\_2)$.


What I want to prove (as part of a more general statement) is that if $\|A\|\_{p\_1}<\|B\|\_{p\_1}$ and $\|A\|\_{p\_2}<\|B\|\_{p\_2}$ for some $1 \leq p\_1 < p\_2 \leq \infty$, then $\|A\|\_q<\|B\|\_q$ for any $p\_1\leq q\leq p\_2$. 




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I have a proof using interpolation theory, more specifically the Riesz-Thorin interpolation theorem, by defining a linear operator $T$ with $T(B)=A$. I can elaborate on this proof if someone is interested in it, but it seems to me like there should be a simpler way to show this, as this theorem talks about much more general spaces. 


I wasn't able to find anything about this question in the literature, but maybe I just don't know the right keywords. My background is more in discrete mathematics, so it is possible I'm just overlooking something obvious here, but I asked some people in functional analysis and they also (offhand) didn't know a simpler way.



Is there an elementary proof for preserving inequalities under the change of l_p metrics?
The inequality is not true in higher dimensions.
For instance, let $A$ have two components equal to 1/2 and 200 components eqaul to 1/100, and the rest zeros, and let $B$ have ten components equal to 1/4 and the rest zeros.
We compute
$$\|A\|\_1=3,\|B\|\_1=5/2,\|A\|\_2=0.7211,\|B\|\_2=0.7906,\|A\|\_3=0.6301,\|B\|\_3=0.5386.$$
Hence
$$\|A\|\_1>\|B\|\_1, \|A\|\_3>\|B\|\_3,$$
but 
$$\|A\|\_2<\|B\|\_2.$$


