Convex-profile inversion, introduced by S.J. Ostro and R. Connelly (1984, Icarus 57, 443-463), is reformulated, extended, and calibrated as a theory for physical interpretation of an asteroid's lightcurve and then is used to constrain the shapes of selected asteroids. Under ideal conditions, one can obtain the asteroid's "mean cross section" C, a convex profile equal to the average of the convex envelopes of all the surface contours parallel to the equatorial plane. C is a unique, rigorously defined, two-dimensional average of the three-dimensional shape and constitutes optimal extraction of shape information from a nonopposition lightcurve. For a lightcurve obtained at opposition, C' s odd harmonics are not accessible, but one can estimate the asteroid's symmetrized mean cross section, Cs. To convey visually the physical meaning of the mean cross sections, we show C and Cs calculated numerically for a regular convex shape and for an irregular, nonconvex shape. The ideal conditions for estimating C from a lightcurve are Condition GEO, that the scattering is uniform and geometric; Condition EVIG, that the viewing-illumination geometry is equatorial; Condition CONVEX, that all of the asteroid's surface contours parallel to the equatorial plane are convex; and Condition PHASE, that the solar phase angle θ ≠ 0. Condition CONVEX is irrelevant for estimation of Cs. Definition of a rotation- and scale-invariant measure of the distance, Ω, between two profiles permits quantitative comparison of the profile's shapes. Useful descriptors of a profile's shape are its noncircularity, Ω c, defined as the distance of the profile from a circle, and the ratio β ∗ of the maximum and minimum values of the profile's breath function β(θ), where θ is rotational phase. C and Cs have identical breadth functions. At opposition, under Conditions EVIG and GEO, (i) the lightcurve is equal to C' s breadth function, and (ii) β ∗ = 10 0.4Δm, where Δm is the lightcurve peak-to-valley amplitude in magnitudes; this is the only situation where Δm has a unique physical interpretation. An estimate ? can be distorted if the applicable ideal conditions are violated. We present results of simulations designed to calibrate the nature, severity, and predictability of such systematic error. Distortion introduced by violation of Condition EVIG depends on θ, on the asteroid-centered declinations δs and δE of the Sun and Earth, and on the asteroid's three-dimensional shape. Violations of Condition EVIG on the order of 10° appear to have little effect for convex, axisymmetric shapes. Errors arising from violation of Condition GEO have been studied by generating lightcurves for model asteroids having known mean cross sections and obeying Hapke's photometric function, inverting the lightcurves, and comparing ?. The distance of ? from C depends on θ and rarely is negligible, but values of Ω c and β ∗ for ? resemble those of C rather closely for a range of solar phase angles ( θ ∼ 20° ± 10°) generally accessible for most asteroids. Opportunities for reliable estimation of Cs far outnumber those for C. We have examined how lightcurve noise and rotational-phase sampling rate propagate into statistical error in ? and offer guidelines for acquisition of lightcurves targeted for convex-profile inversion. Estimates of C and/or Cs are presented for selected asteroids, along with profile shape descriptors and goodness-of-fit statistics for the inverted lightcurves. For 15 Eunomia and 19 Fortuna, we calculate weighted estimates of C from lightcurves taken at different solar phase angles.