In this thesis we study convex subcomplexes of spherical buildings. In particular, we are interested in a question of J. Tits which goes back to the 50’s, the so-called Center

Conjecture. It states that a convex subcomplex of a spherical building is a subbuilding or the building automorphisms preserving the subcomplex have a common fixed point in it.

A proof of the Center Conjecture for the buildings of classical types (An, Bn and Dn)

has been given by B. Muehlherr and J. Tits in [MT06]. The F4-case was presented by C.

Parker and K. Tent in a talk in Oberwolfach [PT08]. Both approaches use combinatorial methods from incidence geometry. B. Leeb and the author gave in [LR09] differentialgeometric proofs for the cases F4 and E6 from the point of view of the theory of metric spaces with curvature bounded from above.

In this work we develop the differential-geometric approach further. Our main result is the proof of the Center Conjecture for buildings of type E7 and E8, whose geometry

is considerably more complicated. In particular, this completes the proof of the Center Conjecture for all thick spherical buildings. We also give a short differential-geometric proof for the classical types. Finally, we show how the cases F4, E6 and E7 can be deduced

from the E8-case.