Minkowski's Inequality

Lemma

is convex for .

Proof

Need to show

WLOG . It is certainly true for (then equality holds).
Now take derivative and show LHS>RHS.

Minkowski’s Inequality

Let and (see L norms).
Then and

Proof

First show if then for all .
For each have (using lemma)
Now sum over and get that RHS is at most 1, so LHS is at most 1.
Now for general ,
apply the inequality to and
and take
and get the result.