Question 8
$S$ is a sphere with unit radius. $P$ is a plane through the center. For any point $x$ on the sphere, $f(x)$ is the perpendicular distance from $x$ to $P$. Show that if $x$, $y$, $z$ are the ends of three mutually perpendicular radii, then $f(x)^2 + f(y)^2 + f(z)^2 = 1$ (*).
Now let $g(x)$ be any function on the points of $S$ taking non-negative real values and satisfying (*). Regard the intersection of $P$ and $S$ as the equator, the poles as the points with $f(x) = 1$ and lines of longitude as semicircles through both poles.
(1) If $x$ and $y$ have the same longitude and both lie on the same side of the equator with $x$ closer to the pole, show that $g(x) \gt g(y)$.
(2) Show that for any $x$, $y$ on the same side of the equator with $x$ closer to the pole than $y$ we have $g(x) \gt g(y)$.
(3) Show that if $x$ and $y$ are the same distance from the pole then $g(x) = g(y)$.
(4) Show that $g(x) = f(x)$ for all $x$.
Now let $g(x)$ be any function on the points of $S$ taking non-negative real values and satisfying (*). Regard the intersection of $P$ and $S$ as the equator, the poles as the points with $f(x) = 1$ and lines of longitude as semicircles through both poles.
(1) If $x$ and $y$ have the same longitude and both lie on the same side of the equator with $x$ closer to the pole, show that $g(x) \gt g(y)$.
(2) Show that for any $x$, $y$ on the same side of the equator with $x$ closer to the pole than $y$ we have $g(x) \gt g(y)$.
(3) Show that if $x$ and $y$ are the same distance from the pole then $g(x) = g(y)$.
(4) Show that $g(x) = f(x)$ for all $x$.