= Fixed point of a finite Euclidean isometry group
A finite <group> of <Euclidean isometries> has a common <fixed point>: average any orbit. Each <isometry> is an <affine map> and therefore preserves this finite affine average; applying a <group> element merely permutes the orbit terms. To see that an isometry is affine, subtract its value at zero and use the <polarization identity> to show that the resulting map preserves inner products. Images of a standard <orthonormal basis> then determine a linear orthogonal map.
Back to article page
