  An upper bound on the capacity of multiple-input multiple-output (MIMO)
Gaussian fading channels is derived under peak amplitude constraints. The upper
bound is obtained borrowing concepts from convex geometry and it extends to
MIMO channels notable results from the geometric analysis on the capacity of
scalar Gaussian channels. Relying on a sphere packing argument and on the
renowned Steiner's formula, the proposed upper bound depends on the intrinsic
volumes of the constraint region, i.e., functionals defining a measure of the
geometric features of a convex body. The tightness of the bound is investigated
at high signal-to-noise ratio (SNR) for any arbitrary convex amplitude
constraint region, for any channel matrix realization, and any dimension of the
MIMO system. In addition, two variants of the upper bound are proposed: one is
useful to ensure the feasibility in the evaluation of the bound and the other
to improve the bound's performance in the low SNR regime. Finally, the upper
bound is specialized for two practical transmitter configurations, either
employing a single power amplifier for all transmitting antennas or a power
amplifier for each antenna.
