Volume concentrates near the boundary

NumericalBeginner⏱ 15 minHigh-Dim Probabilityvolumeballshellhigh dimension

Problem

What fraction of the volume of the unit ball in $\R^d$ lies at radius less than $1-\eps$ ?
Find the $\eps$ for which that fraction is $1\%$ , as a function of $d$ , and evaluate it for $d = 3, 100, 1000$ .
What fraction of the volume of the unit cube $[-1,1]^d$ lies inside the inscribed ball? Evaluate for $d = 2,10,100$ and comment.

Compute the requested quantities and say what they mean. A number without an interpretation is not an answer.

Hints — 0 of 2 revealed

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

Problem	Field	Type	Difficulty	Time
Simulate the measure of a spherical cap run	High-Dim Probability	Coding	Intermediate	30 min
The volumetric lower bound on covering numbers	High-Dim Probability	Proof	Intermediate	30 min
Concentration of the norm — the transfer step	High-Dim Probability	Proof	Advanced	30 min
Distance contrast in high dimension	High-Dim Probability	Proof	Advanced	1 hour
How many nearly-orthogonal directions fit in R^d?	High-Dim Probability	Proof	Advanced	30 min