Variation diminishing property

The variation diminishing property of Bézier curves is that they are smoother than the polygon formed by their control points.

[1] This property was first studied by Isaac Jacob Schoenberg in his 1930 paper, Über variationsvermindernde lineare Transformationen.

[3] The proof uses the process of repeated degree elevation of Bézier curve.

The process of degree elevation for Bézier curves can be considered an instance of piecewise linear interpolation.

The blocks of Dirichlet-to-Neumann maps of planar graphs have the variation diminishing property.