MathDB

Suppose that the closed subset

K

of the sphere

S^2=\{ (x,y,z)\in \mathbb{R}^3\colon x^2+y^2+z^2=1 \}

is symmetric with respect to the origin and separates any two antipodal points in

S^2 \backslash K

. Prove that for any positive

\varepsilon

there exists a homogeneous polynomial

P

of odd degree such that the Hausdorff distance between

Z(P)=\{ (x,y,z)\in S^2 \colon P(x,y,z)=0\}

and

K

is less than

\varepsilon

Problem Statement