Two points in a thin spherical shell are joined by a curve shorter than the diameter of the shell. Show that the curve lies entirely in one hemisphere.
Solution
Suppose the shell has diameter 2. Let M be the midpoint of the curve. Let O be the center of the shell and X the midpoint of MO. Let S be the circle center X radius √3)/2 in the plane normal to OM. Then S lies in the shell and every point of S is a distance (in space) of 1 from M. Hence the curve cannot cross S (because if it crossed at Y, we would have SY ≥ 1 along the curve, but the curve has length < 2, so SY < 1. So we have a stronger result than required.
© John Scholes
jscholes@kalva.demon.co.uk
19 Aug 2002