P-norms for p>2, p<∞

A few weeks ago I posted a fun Math/DS question on Twitter that got some really great replies. Figured I’d repost here in case any of you have your own favorite answers.

Here’s the tweet:

And here’s the question:

Consider the Lp norm for p>2 but not the infinity norm. Tell me p and an application space where this norm is right/good/useful/captures some important structure.

Some of my favorite replies:

What about yours? Any examples you find especially interesting or beautiful? Have you used any of the above in a real application?