  Context. The recent estimates of the 3D shape of the M/Xe-type triple
asteroid system (216) Kleopatra indicated a density of 5 g.cm$^{-3}$. Such a
high density implies a high metal content and a low porosity which is not easy
to reconcile with its peculiar dumbbell shape. Aims. Given the unprecedented
angular resolution of the VLT/SPHERE/ZIMPOL camera, we aim to constrain the
mass and the shape of Kleopatra with high accuracy, hence its density. Methods.
We combined our new VLT/SPHERE observations of Kleopatra recorded in 2017 and
2018 with archival data, as well as lightcurve, occultation, and delay-Doppler
images, to derive its 3D shape model using two different algorithms (ADAM,
MPCD). Furthermore, an N-body dynamical model allowed us to retrieve the
orbital elements of the two moons as explained in the accompanying paper.
Results. The shape of Kleopatra is very close to an equilibrium dumbbell figure
with two lobes and a thick neck. Its volume equivalent diameter
(118.75$\pm$1.40) km and mass (2.97$\pm$0.32) 10$^{18}$ kg imply a bulk density
of (3.38$\pm$0.50) g cm$^{-3}$. Such a low density for a supposedly metal-rich
body indicates a substantial porosity within the primary. This porous structure
along with its near-equilibrium shape is compatible with a formation scenario
including a giant impact followed by reaccumulation. Kleopatra's current
rotation period and dumbbell shape imply that it is in a critically rotating
state. The low effective gravity along the equator of the body, together with
the equatorial orbits of the moons and possibly rubble-pile structure, opens
the possibility that the moons formed via mass shedding. Conclusions. Kleopatra
is a puzzling multiple system due to the unique characteristics of the primary.
It deserves particular attention in the future, with the Extremely Large
Telescopes and possibly a dedicated space mission.
