Are there well-known or somewhere-well-studied methods of interpolation that will produce convex approximants when fed data that could conceivably fit a convex function? A lot of stuff (e.g. Lagrange-style polynomial interpolation, piecewise-smooth polynomial interpolation, etc.) seems not to be guaranteed to produce a result that matches this feature of the input data.
Here the input conditions would be a finite set of ordered real pairs $(x,y)$ satisfying divided-difference conditions compatible with something smooth and convex matching the input, and the desired output (whether an exact match, or only a close match, I truly don't care, although it would be ideal if the function fit the points exactly) would be a smooth function that, whatever else it might do, is convex, real-valued and defined at least on some interval including all of the $x$-values, and either exactly or tries its best to represent the input data.
I do apologize if this is a vague question, but I do not have much more context than this. I am working on shedding light on an applied problem where it would be desirable to refer a reader to methods of interpolation of finite sets of ordered pairs of points in the plane, which, when these points satisfy conditions that would not forbid a convex function from matching them, actually select a convex function that matches them or best approximates them in some sense. I feel like this ought to be something people have studied in detail and I just simply don't know of any references.
