So here's the Hlawka's inequality where $\alpha,\beta,\gamma$ belong to an inner product space:
$\|\alpha + \beta\| + \|\beta + \gamma\| + \|\gamma + \alpha\| \leq \|\alpha\| + \|\beta\| + \|\gamma\| + \|\alpha + \beta + \gamma\|$
I know the original proof as is stated in Analytic Inequalities by D.S. Mitrinović. We multiply it by the right side and after simplifying it and using triangle inequality, it is done.
My first idea for proving this inequality was to square both sides, and regroup the result and then prove the inequality for each group. something like the second proof in here.
That proof only holds for complex number, but the author has said that it can be generalized to vectors in an inner product space.
So that's where I'm stuck. It is easy for complex numbers. But I couldn't get any further on the general case.
In summary I want to show that the following inequality holds:
$
\|\alpha + \beta\|.\|\gamma + \beta\| \leq \|\alpha\|.\|\gamma\| + \|\beta\|.\|\alpha + \gamma + \beta\|
$
2026-04-11 16:49:00.1775926140
Hlawka's inequality
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