I've always found the usual proofs of both Minkowski's and Hölder's inequalities extremely unintuitive and unsatisfying. It is very unclear how someone who wanted to prove Minkowski's inequality would come up with Hölder's inequality and the tricks used to prove them. Due to this, I've came up myself with an alternative proof that holds in $\mathbb R^n$. By classifying all possible norms in these spaces geometrically, it becomes simple to prove the special case of Minkowski's inequality in this case. Even though it takes much more work, I find this much more satisfying, because every step is clear and motivated.
Are there more intuitive ways to prove Minkowski's and/or Hölder's inequalities in full generality? Or, alternatively, is there some natural intutive route to follow that would lead us to the usual proofs of them?
