  This work presents an interferometric study of the massive-binary fraction in
the Orion Trapezium Cluster with the recently comissioned GRAVITY instrument.
We observe a total of 16 stars of mainly OB spectral type. We find three
previously unknown companions for $\theta ^1$ Ori B, $\theta ^2$ Ori B, and
$\theta ^2$ Ori C. We determine a separation for the previously suspected
companion of NU Ori. We confirm four companions for $\theta ^1$ Ori A, $\theta
^1$ Ori C, $\theta ^1$ Ori D, and $\theta ^2$ Ori A, all with substantially
improved astrometry and photometric mass estimates. We refine the orbit of the
eccentric high-mass binary $\theta ^1$ Ori C and we are able to derive a new
orbit for $\theta ^1$ Ori D. We find a system mass of 21.7 $M_{\odot}$ and a
period of $53$ days. Together with other previously detected companions seen in
spectroscopy or direct imaging, eleven of the 16 high-mass stars are multiple
systems. We obtain a total number of 22 companions with separations up to 600
AU. The companion fraction of the early B and O stars in our sample is about 2,
significantly higher than in earlier studies of mostly OB associations. The
separation distribution hints towards a bimodality. Such a bimodality has been
previously found in A stars, but rarely in OB binaries, which up to this point
have been assumed to be mostly compact with a tail of wider companions. We also
do not find a substantial population of equal-mass binaries. The observed
distribution of mass ratios declines steeply with mass, and like the direct
star counts, indicates that our companions follow a standard power law initial
mass function. Again, this is in contrast to earlier findings of flat mass
ratio distributions in OB associations. We exclude collision as a dominant
formation mechanism but find no clear preference for core accretion or
competitive accretion.
