We study a multidimensional collective decision under incomplete information. Agents have Euclidean preferences and vote by simple majority on each issue (dimension), yielding the coordinate‐wise median. Judicious rotations of the orthogonal axes—the issues that are voted upon—lead to...

We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems. Our main results show that each extreme...