The Center Conjecture for spherical buildings of types F4 and E6

We prove that a convex subcomplex of a spherical building of type F4 or E6 is a subbuilding or the automorphisms of the subcomplex fix a point on it. Our approach is differential-geometric and based on the theory of metric spaces with curvature bounded above. We use these techniques also to give another proof of the same result for the spherical buildings of classical type.