Are convex fractals possible?

Question

Is it possible to have a fractal shape (and by that I mean a shape with non-integer Minkowski dimension) that is convex? I'm pretty sure it isn't, as every example of a fractal I can think of is concave; that might be a lack of knowledge or imagination on my part, and I have no idea how you would approach a rigorous proof. Any tips?

Answer 1

The key idea is that convex sets either have nonempty interior or are contained in a lower dimensional hyperplane (see for example here Convex Set with Empty Interior Lies in an Affine Set). This means we can reduce the dimension until our set has nonempty interior. However, if your sets contains an open set it cannot be a fractal (see for example here Hausdorff dimension of an open set).