Binary operations in spherical convex geometry (Q2833070)

Binary operations between convex bodies on the \(n\)-dimensional Euclidean unit sphere are studied, focusing on those operations that are covariant under projections onto great sub-spheres. Necessary and sufficient conditions for a binary operation between proper convex bodies to be projection covariant and continuous with respect to the Hausdorff metric are proved. The operations satisfying these conditions are called trivial in this paper. The main result shows that the convex hull is the only non-trivial projection covariant operation in the above described context. The case of binary operations between convex bodies in a fixed open hemisphere that are projection covariant with respect to the center of the hemisphere is discussed, putting them into a one-to-one correspondence with projection covariant on convex bodies in \(\mathbb{R}^n\).