# 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

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.

• For the first problem, consider take power $p$ on both sides. Apr 16, 2021 at 10:49

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 $$\varphi(a)=\varphi(a)-\varphi(0)=\cdots\geq 0$$.