  We investigate and prove Lieb-Oxford bounds in one dimension by studying
convex potentials that approximate the ill-defined Coulomb potential. A
Lieb-Oxford inequality establishes a bound of the indirect interaction energy
for electrons in terms of the one-body particle density $\rho_\psi$ of a wave
function $\psi$. Our results include modified soft Coulomb potential and
regularized Coulomb potential. For these potentials, we establish
Lieb-Oxford-type bounds utilizing logarithmic expressions of the particle
density. Furthermore, a previous conjectured form $I_\mathrm{xc}(\psi)\geq -
C_1 \int_{\mathbb R} \rho_\psi(x)^{2} \mathrm{d}x$ is discussed for different
convex potentials.
