Show that $(\sum_{i=1}^n |x_i - y_i|^p)^\frac{1}{p} \leq \sum_{i=1}^n |x_i - y_i|$ for all x,y in $\mathbb{R}^n$ using Minkowski's Inequality

Question

I am currently studying Metric Spaces and the following problem arose as a sub-problem of an exercise I am working through:

It is true that $(\sum_{i=1}^n |x_i - y_i|^p)^\frac{1}{p} \leq \sum_{i=1}^n |x_i - y_i|$ for all x,y in $\mathbb{R}^n$ where $p > 1$. This follows from Minkowski's inequality.

I have also considered the related version:

Conjecture: $(\sum_{i=1}^n |x_i - y_i|^p)^\frac{1}{p} \leq C\sum_{i=1}^n |x_i - y_i|$ for all x,y in $\mathbb{R}^n$ where $p > 1$ for an appropriate positive real constant $C$.

However I am quite stuck and have not made progress on both of them. If anyone can provide a hint that uses Minkowski's Inequality to prove one of the above statements or a counterexample, that would be greatly appreciated.

Answer 1

So did you already establish the fact that $a^{p}+b^{p}\leq(a+b)^{p}$ for $a,b\geq 0$?

If so, then it is only the induction matter to obtain the result: $|a_{1}|^{p}+(|a_{2}|^{p}+\cdots+|a_{n}|^{p})\leq|a_{1}|^{p}+(|a_{2}|+\cdots+|a_{n}|)^{p}\leq(|a_{1}|+(|a_{2}|+\cdots+|a_{n}|))^{p}$.

So it is all about to show that $a^{p}+b^{p}\leq(a+b)^{p}$. To see this, by substituting $a/b$, one needs to show only that $1+a^{p}\leq(a+1)^{p}$. This can be done by an application of Mean Value to the function $\varphi(a)=(a+1)^{p}-a^{p}-1$ that $\varphi(a)=\varphi(a)-\varphi(0)=\cdots\geq 0$.