When does the equality hold for norm equivalence

Question

We know that for a vector $x\in \mathbb{R}^n$, its 1-norm and 2-norm satisfy that $$\frac{1}{n}\|x\|_1\le\|x\|_2\le \|x\|_1,$$ could anyone please give me some hints that on what condition these equality holds?

Answer 1

The null vector $x=\vec{0}_n$ yields the equality of norms.

Answer 2

Hint: If you are given a proof for those estimates, look at them. The proofs contain some other inequalities that you may be more familiar with. When does equality hold in them?

A different kind of hint: What special kind of vectors can you come up with in $\mathbb R^n$? What are some simple nonzero vectors?

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The first inequality is not optimal and thus never reached. If you can show that $\frac{1}{\sqrt n}\|x\|_1\le\|x\|_2$, you can conclude that equality only holds for $x=0$ (or when $n=1$). To prove this, find a vector $y$ so that $x\cdot y=\|x\|_1$ and use Cauchy-Schwarz.