1 6 1 The Strange Geometry Of High Dimensional Spaces

Since $\|v\|^2=\sum_{i=1}^p \|e_i\|^2=1+\ldots+1=p$ by pythagorean theorem and $\langle e_i,v\rangle=1$ we have

(1)

\begin{align} \cos(\theta_i)=\frac{\langle e_i,v\rangle}{\|e_i\|\|v\|}=\frac{1}{\sqrt{p}}, \end{align}

which tends to 0 when $p$ goes to infinity.

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Here is a suggestion for a related exercise:

Let $V$ be a random vector in dimension $p$ whose coordinates are i.i.d. random variables with distribution symmetric around 0 and having no atom in 0 (no integrability conditions). Let $V'$ be an independent copy of $V$ and denote $\theta$ the random angle of these two vectors.

Establish that $E[\cos (\theta)]=0$ and $\mathrm{Var}[\cos(\theta)] = \frac{1}{p}$, implying that $V,V'$ tend to be close to orthogonal with probability increasing with the dimension.