Every convex homogeneous polynomial (or form) is nonnegative. A natural question, raised by Parrilo, is whether every convex form has the stronger property of being a sum of squares. Blekherman subsequently showed that convex forms that are not sums of squares exist via a nonconstructive argument, but until now no explicit examples seem to be known. In this talk I will discuss an explicit example of a convex form of degree four in 272 variables that is not a sum of squares. The form is related to the Cauchy-Schwarz inequality over the octonions. I will also discuss connections between this question and the quality of sum-of-squares-based relaxations of polynomial optimization problems over the sphere.