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An investigation is launched into the fundamental characteristics of operations on and between sets, with a focus on compact convex sets and star sets (compact sets star-shaped with respect to the origin) in $n$-dimensional Euclidean space ${ℝ}^n$. It is proved that if $n\ge 2$, with three trivial exceptions, an operation between origin-symmetric compact convex sets is continuous in the Hausdorff metric, $GL\left(n\right)$ covariant, and associative if and only if it is $L_p$ addition for some $1\le p\le \infty$. It is also demonstrated that if $n\ge 2$,...