  The axion emerges in extensions of the Standard Model that explain the
absence of CP violation in the strong interactions. Simultaneously, it can
provide naturally the cold dark matter in our universe. Several searches for
axions and axion-like particles (ALPs) have constrained the corresponding
parameter space over the last decades but no unambiguous hints of their
existence have been found. The axion mass range below 1 meV remains highly
attractive and a well motivated region for dark matter axions. In this White
Paper we present a description of a new experiment based on the concept of a
dielectric haloscope for the direct search of dark matter axions in the mass
range of 40 to 400 $\mu$eV. This MAgnetized Disk and Mirror Axion eXperiment
(MADMAX) will consist of several parallel dielectric disks, which are placed in
a strong magnetic field and with adjustable separations. This setting is
expected to allow for an observable emission of axion induced electromagnetic
waves at a frequency between 10 to 100 GHz corresponding to the axion mass.
