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Let $f(s)$ stand for the difference between the left- and right-hand sides of your displayed inequality. We want to show that $f<0$ on the interval $(0,1/ \sqrt3)$. Let $$f_1(s):=f'(s)\frac{8 \sqrt{1- …

Yes, this follows immediately from the triangle inequality on the sphere -- cf. e.g. this or this.

The answer is no. E.g., let $P$ be the set $\{1,e^{it},e^{i3t}\}$ of points on the unit circle in $\mathbb C=\mathbb R^2$, where $t$ is a small positive real number. Then the geodesic mean of $P$ is $ …

$\newcommand\al\alpha$By spherical symmetry, the conditional distribution of $\|X-Y\,\sqrt{1-\al}\|$ given $Y$ does not depend on $Y$. So, letting $e_1:=(1,0,\dots,0)$ and writing $X=(X_1,\dots,X_d)$, …

$\newcommand\al\alpha\newcommand\be\beta\newcommand\ga\gamma$It appears that the question is as follows: Given unit vectors $a,b,c$ with angles $$\al:=\cos^{-1}(b\cdot c),\quad \be:=\cos^{-1}(a\cdot c …

$\newcommand{\R}{\mathbb R}\newcommand{\SSS}{\mathbb S}\newcommand{\tS}{\tilde S}\newcommand{\tR}{\tilde R}$This is to detail and formalize the nice counterexample outlined in fedja's comment. First, …