  Constraining Jupiter's internal structure is crucial for understanding its
formation and evolution history. Recent interior models of Jupiter that fit
Juno's measured gravitational field suggest an inhomogeneous interior and
potentially the existence of a diluted core. These models, however, strongly
depend on the model assumptions and the equations of state used. A
complementary modelling approach is to use empirical structure models. These
can later be used to reveal new insights on the planetary interior and be
compared to standard models. Here we present empirical structure models of
Jupiter where the density profile is constructed by piecewise-polytropic
equations. With these models we investigate the relation between the normalized
moment of inertia (MoI) and the gravitational moments $J_2$ and $J_4$. Given
that only the first few gravitational moments of Jupiter are measured with high
precision, we show that an accurate and independent measurement of the MoI
value could be used to further constrain Jupiter's interior. An independent
measurement of the MoI with an accuracy better than $\sim 0.1\%$ could
constrain Jupiter's core region and density discontinuities in its envelope. We
find that models with a density discontinuity at $\sim$ 1 Mbar, as would
produce a presumed hydrogen-helium separation, correspond to a fuzzy core in
Jupiter. We next test the appropriateness of using polytropes, by comparing
them with empirical models based on polynomials. We conclude that both
representations result in similar density profiles and ranges of values for
quantities like core mass and MoI.
