Concentration of the norm — the transfer step

ProofAdvanced⏱ 30 minHigh-Dim Probabilitynorm concentrationtransferelementary inequality

Theory: High-Dimensional Geometry and Concentration of the Norm →

Problem

Prove the elementary inequality $|z-1| \le |z^2-1|$ for all $z\ge0$ , and explain precisely how it converts a concentration bound for $\|X\|_2^2/d$ around $1$ into a concentration bound for $\|X\|_2$ around $\sqrt d$ with no dimension-dependent loss.

Write a complete argument. Every inequality you use should be one you could state precisely.

Hints

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Solution

Opening this before you have a real attempt on paper is the fastest way to feel like you understand something you cannot reproduce.

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