In this paper we characterize the degenerate elliptic equations <b>F</b>(D<sup>2</sup>u) = 0 whose subsolutions (<b>F</b>(D<sup>2</sup>u) &ge; 0) satisfy the strong maximum principle. We introduce an easily computed function f on (0,&infin;) which is determined by <b>F</b>, and we show that the strong maximum principle holds depending on whether &int;<sub> 0<sup>+</sup></sub> dy/f(y) is infinite or finite. This is in the spirit of previous work characterizing the ordinary maximum principle in terms of the geometry of the set of symmetric matrices 
F = {<b>F</b> &ge; 0}. Along the way, radial subsolutions are characterized, and, as an application, a sufficient condition for strong comparison is established. A number of examples, important for the theory of such equations, are examined.