  A comprehensive study of the application of SO$(D+1)$ coherent states of
Perelomov type to loop quantum gravity in general spacetime dimensions $D+1\geq
3$ is given in this paper. We focus on so-called simple representations of
SO$(D+1)$ which solve the simplicity constraint and the associated homogeneous
harmonic function spaces. With the harmonic function formulation, we study
general properties of the coherent states such as the peakedness properties and
the inner product. We also discuss the properties of geometric operators
evaluated in the coherent states. In particular, we calculate the expectation
value of the volume operator, and the results agree with the ones obtained from
the classical label of the coherent states up to error terms which vanish in
the limit of large representation labels $N$, i.e. the analogue of the large
spin limit in standard $3+1$-dimensional loop quantum gravity.
