  We consider Brownian motion under resetting in higher dimensions for the case
when the return of the particle to the origin occurs at a constant speed. We
investigate the behavior of the probability density function (PDF) and of the
mean-squared displacement (MSD) in this process. We study two different
resetting protocols: exponentially distributed time intervals between the
resetting events (Poissonian resetting) and resetting at fixed time intervals
(deterministic resetting). We moreover discuss a general problem of the
invariance of the PDF with respect to the return speed, as observed in the
one-dimensional system for Poissonian resetting, and show, that this one
dimensional situation is the only one in which such an invariance can be found.
However, the invariance of the MSD can still be observed in higher dimensions.
