  Early observations of supernova light curves are powerful tools for shedding
light on the pre-explosion structures of their progenitors and their mass-loss
histories just prior to explosion. Some core-collapse supernovae that are
detected during the first days after the explosion prominently show two peaks
in the optical bands, including the $R$ and $I$ bands, where the first peak
appears to be powered by the cooling of shocked surface material and the second
peak is clearly powered by radioactive decay. Such light curves have been
explored in detail theoretically for SN 1993J and 2011dh, where it was found
that they may be explained by progenitors with extended, low-mass envelopes.
Here we generalize these results. We explore, first, whether any double-peaked
light curve of this type can be generated by a progenitor with a "standard"
density profile, such as a red supergiant or a Wolf-Rayet star. We show that a
standard progenitor (1) cannot produce a double-peaked light curve in the $R$
and $I$ bands, and (2) cannot exhibit a fast drop in the bolometric luminosity
as is seen after the first peak. We then explore the signature of a progenitor
with a compact core surrounded by extended, low-mass material. This may be a
hydrostatic low-mass envelope or material ejected just prior to the explosion.
We show that it naturally produces both of these features. We use this result
to provide simple formulae to estimate (1) the mass of the extended material
from the time of the first peak, (2) the extended material radius from the
luminosity of the first peak, and (3) an upper limit on the core radius from
the luminosity minimum between the two peaks.
