  Based on new observations and improved modeling techniques, we have
reanalyzed seven Cepheids in the Large Magellanic Cloud. Improved physical
parameters have been determined for the exotic system OGLE LMC-CEP-1718
composed of two first-overtone Cepheids and a completely new model was obtained
for the OGLE LMC-CEP-1812 classical Cepheid. This is now the shortest period
Cepheid for which the projection factor is measured. The typical accuracy of
our dynamical masses and radii determinations is 1%. The radii of the six
classical Cepheids follow period--radius relations in the literature. Our very
accurate physical parameter measurements allow us to calculate a purely
empirical, tight period--mass--radius relation that agrees well with
theoretical relations derived from non-canonical models. This empirical
relation is a powerful tool to calculate accurate masses for single Cepheids
for which precise radii can be obtained from Baade--Wesselink-type analyses.
The mass of the type-II Cepheid $\kappa$ Pav, $0.56 \pm 0.08 M_\odot$,
determined using this relation is in a very good agreement with theoretical
predictions. We find large differences between the p-factor values derived for
the Cepheids in our sample. Evidence is presented that a simple
period--p-factor relation shows an intrinsic dispersion, hinting at the
relevance of other parameters, such as the masses, radii, and radial velocity
variation amplitudes. We also find evidence that the systematic blueshift
exhibited by Cepheids, is primarily correlated with their gravity. The
companion star of the Cepheid in the OGLE LMC-CEP-4506 system has a very
similar temperature and luminosity, and is clearly located inside the Cepheid
instability strip, yet it is not pulsating.
