Recently I read some course notes and articles on Bézier curves. They all sum up the properties of Bézier curves, like the partition-of-unity property of the basis functions (Bernstein polynomials), variation diminishing property of the curve (the curve doesn't wiggle/oscillate more than the control polygon does), and also the convex hull property.
Apparently the latter is an important property, but I cannot find why this would be the case. It has something to do with the numerical stability of convex combinations -- but then again, why are convex combinations more "numerically stable" than other types of combinations, like affine ones?
So, when one compares a Bézier curve to e.g. a Lagrange curve (which interpolates all its control points), the former curve remains inside the polygon spanned by its control points (the convex hull), while the latter doesn't. I know this is because a Bézier curve is created using only convex combinations while the Lagrange curve isn't. But why is this property so important? What makes it better than a Lagrange curve?