An algebraic unit all of whose algebraic conjugates other than itself lie on or inside unit circle. In 1933, Lehmer observed that the smallest Salem number in each degree gets steadily smaller until 10, where the minimum achieved is now called Lehmer's number, and then starts to go up again as the degree grows. In the first half of the talk I'll discuss the minimization problem for Salem numbers, and other related questions about the "size" of algebraic integers. In the second half I will connect the minimization problems to questions about Coxeter graphs and their generalizations.