in the book "CAGD" by Farin he gives a proof of the variation diminishing property for piecewise linear interpolation. Given a continuous curve $c$ in $\mathbb{R}^3$ we define a piecewise linear interpolant of the curve $\mathcal{PL}\,c$ as a polygon (sequence of straight line segments joining points $\mathbb{b}_i,\mathbb{b}_{i+1}$ whose vertices lie on the curve. The variation diminishing property states that, given an arbitrary plane, the number of intersections of the plane with the curve is less or equal to the number of intersections of the plane with the piecewise linear interpolant.
I don't get what happens when the plane contains one of the straight line segments of $\mathcal{PL}\,c$. How precisely is the defined this "number of intersections"?