Steiner symmetrization

Steiner symmetrization replaces every chord perpendicular to a chosen axis by a centered chord of the same length.

Measurable planar sets with equal one-dimensional slice lengths in one direction have equal areas.

Every perpendicular chord keeps its length under Steiner symmetrization, so Fubini's theorem preserves the total area.

For a convex planar domain between graphs $u_{-}$ and $u_{+}$, the Euclidean triangle inequality giveswhich is the pointwise perimeter comparison with its symmetrization.

Equality holds exactly when $u_{+}^{\prime }=-u_{-}^{\prime }$, so the midpoints of all perpendicular chords lie on one line parallel to the symmetrizing axis.

If Steiner symmetrization preserves area and cannot lower the perimeter of a minimizer, equality forces an axis of symmetry in the chosen direction. Applying every direction gives symmetry axes in all directions.