A family of repulsive neutral conductor geometries via abstract vector spaces

Recently it was shown that it is possible for a neutral, isolated conductor to repel a point charge (or, a point dipole). Here we prove this fact using general properties of vectors and operators in an inner-product space. We find that a family of neutral, isolated conducting surface geometries, whose shape lies somewhere between a hemispherical bowl and an ovoid, will repel a point charge. In addition, we find another family of surfaces (with a different shape) that will repel a point dipole. The latter geometry can lead to Casimir repulsion.