  In their celebrated work, B. Andrews and J. Clutterbuck proved the
fundamental gap (the difference between the first two eigenvalues) conjecture
for convex domains in the Euclidean space and conjectured similar results holds
for spaces with constant sectional curvature. We prove the conjecture for the
sphere. Namely when $D$, the diameter of a convex domain in the unit $S^n$
sphere, is $\le \frac{\pi}{2}$, the gap is greater than the gap of the
corresponding $1$-dim sphere model. We also prove the gap is $\ge
3\frac{\pi^2}{D^2}$ when $n \ge 3$, giving a sharp bound. As in
Andrews-Clutterbuck's proof of the fundamental gap, the key is to prove a super
log-concavity of the first eigenfunction.
