  In this work, we present an approach for the efficient treatment of
parametrized geometries in the context of POD-Galerkin reduced order methods
based on Finite Volume full order approximations. On the contrary to what is
normally done in the framework of finite element reduced order methods,
different geometries are not mapped to a common reference domain: the method
relies on basis functions defined on an average deformed configuration and
makes use of the Discrete Empirical Interpolation Method (D-EIM) to handle
together non-affinity of the parametrization and non-linearities. In the first
numerical example, different mesh motion strategies, based on a Laplacian
smoothing technique and on a Radial Basis Function approach, are analyzed and
compared on a heat transfer problem. Particular attention is devoted to the
role of the non-orthogonal correction. In the second numerical example the
methodology is tested on a geometrically parametrized incompressible
Navier--Stokes problem. In this case, the reduced order model is constructed
following the same segregated approach used at the full order level
