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.