  We present new CO(2-1) observations of a representative sample of 24 local
(z$<$0.02) luminous infrared galaxies (LIRGs) obtained at high spatial
resolution ($<$100 pc) from ALMA. We derive the effective radii of the CO(2-1)
and the 1.3 mm continuum emissions using the curve-of-growth method. LIRGs show
an extremely compact cold molecular gas distribution (median R$_{CO}$ $\sim$0.7
kpc), which is a factor 2 smaller than the ionized gas, and 3.5 times smaller
than the stellar size. The molecular size of LIRGs is similar to that of
early-type galaxies (R$_{CO}\sim$1 kpc) and about a factor of 6 more compact
than local Spirals of similar stellar mass. Only the CO emission in low-z
ULIRGs is more compact than these local LIRGs by a factor of 2. Compared to
high-z (1$<$z$<$6) systems, the stellar sizes and masses of local LIRGs are
similar to those of high-z MS star-forming galaxies (SFG) and about a factor of
2-3 lower than sub-mm galaxies (SMG). The molecular sizes of high-z MS SFGs and
SMGs are larger than those derived for LIRGs by a factor of $\sim$3 and
$\sim$8, respectively. These results indicate that while low-z LIRGs and high-z
MS-SFGs have similar stellar masses and sizes, the regions of current star
formation (traced by the ionized gas) and of potential star-formation (traced
by the molecular gas) are substantially smaller in LIRGs, and constrained to
the central kpc region. High-z galaxies represent a wider population but their
star-forming regions are more extended, even covering the overall size of the
host galaxy. High-z galaxies have larger fractions of gas than low-z LIRGs, and
therefore the formation of stars could be induced by interactions and mergers
in extended disks or filaments with large enough molecular gas surface density
involving physical mechanisms similar to those identified in the central kpc of
LIRGs.
