Perimeter decrease under Steiner symmetrization

For a convex planar domain between graphs

u_{-}

and

u_{+}

, the Euclidean triangle inequality gives

1 + u_{+}^{'}^{2} + 1 + u_{-}^{'}^{2} \geq 4 + (u_{+}^{'} - u_{-}^{'})^{2},

which is the pointwise perimeter comparison with its symmetrization.

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- Equality case in the Steiner perimeter inequality Perimeter decrease under Steiner symmetrization

Equality case in the Steiner perimeter inequality

Equality holds exactly when

u_{+}^{'} = - u_{-}^{'}

, so the midpoints of all perpendicular chords lie on one line parallel to the symmetrizing axis.

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