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

The Center Conjecture for spherical buildings of types F4 and E6