Let where . Show that the argmin of can be written as

The function is strictly convex and . This implies that admits a unique minimizer .

• Case 1: , in which case . This implies . Therefore , which implies . Therefore

from which we deduce . Using again , we deduce and finally,

• Case 2: . In this case, for all , , which is equivalent to . Dividing by and letting , we get which in turn implies . This shows and we therefore have again: