Suppose that the closed subset K of the sphere
S2={(x,y,z)∈R3:x2+y2+z2=1}
is symmetric with respect to the origin and separates any two antipodal points in S2\K. Prove that for any positive ε there exists a homogeneous polynomial P of odd degree such that the Hausdorff distance between
Z(P)={(x,y,z)∈S2:P(x,y,z)=0}
and K is less than ε. college contestsMiklos Schweitzer3D geometryspherepolynomialtopologygeometry