Sums of norm spheres are norm shells and lower triangle inequalities are sharp

TL;DR

This paper explores fundamental properties of norm spheres and shells in normed vector spaces, providing proofs and connecting to recent moment bounds for submartingales, highlighting sharp inequalities.

Contribution

It offers elementary proofs of sums of norm spheres being norm shells and establishes sharp lower triangle inequalities, linking classical analysis with probabilistic bounds.

Findings

Proved that sums of norm spheres are norm shells

Established sharp lower triangle inequalities

Connected norm inequalities to submartingale moment bounds

Abstract

The statements in the title are explained and proved, as a little exercise in elementary normed vector space theory at the level of Chapter 5 of Dieudonn\'e's "Foundations of Mathematical Analysis". A connection to recent moment bounds for submartingales is sketched.