Question

Let S be the set of interior points of a sphere and C be the set of interior points of a circle. Find, with proof, whether there exists a function $f:S\to C$ such that $d(A,B)\le d(f\left(A\right),f\left(B\right)),$ for any points $A,BϵS.$

Answer 1

No such function exists. Indeed, suppose $f:S\to C$ has the enounced property. Consider a cube inscribed in the sphere and assume with no loss of generality that its sides 1. Partition the cube into $n^3$ smaller cubes and let $A_1,A_2...A_{{(n+1)}^3}$ be their vertices. For all $i\ne j$ we have $d(A_i,A_j)\ge \frac{1}{n},$ hence $d(f\left(A_i\right),f\left(A_j\right))\ge \frac{1}{n}.$ It follows that the disks $D_i$ with centers $f\left(A_i\right)$ and radius $\frac{1}{2n}$ are disjoint and contained in a circle C' with radius $r+\frac{1}{n},$ where r is the radius of C. The sum of the areas of these disks is then less than the area of C', hence ${(n+1)}^3\frac{\pi }{4n^2}\le \pi {(r+\frac{1}{n})}^2.$ This inequality cannot hold for sufficiently large n, which proves our claim.