  We introduce a new technique to bound the fluctuations exhibited by a
physical system, based on the Euclidean geometry of the space of observables.
Through a simple unifying argument, we derive a sweeping generalization of
so-called Thermodynamic Uncertainty Relations (TURs). We not only strengthen
the bounds but extend their realm of applicability and in many cases prove
their optimality, without resorting to large deviation theory or
information-theoretic techniques. In particular, we find the best TUR based on
entropy production alone and also derive a novel bound for stationary Markov
processes, which surpasses previous known bounds. Our results derive from the
non-invariance of the system under a symmetry which can be other than time
reversal and thus open a wide new spectrum of applications.
