  The core result of this paper is an upper bound for the ground state energyof
the magnetic Laplacian with constant magnetic field on cones that are contained
in ahalf-space. This bound involves a weighted norm of the magnetic field
related to momentson a plane section of the cone. When the cone is sharp, i.e.
when its section is small, thisupper bound tends to 0. A lower bound on the
essential spectrum is proved for familiesof sharp cones, implying that if the
section is small enough the ground state energy is aneigenvalue. This
circumstance produces corner concentration in the semi-classical limit forthe
magnetic Schr\"odinger operator when such sharp cones are involved.
