Convexity of the adiabatic-connection interaction energy

Prove the conjectured convexity of the interaction energy $W^{\lambda,0}$ as a function of the coupling strength $\lambda$ in density functional theory.

Background

The paper defines $W^{\lambda,\tau}$ as the expectation value of the electron–electron interaction along a family of systems with fixed density, interaction strength $\lambda$, and temperature $\tau$. At zero temperature, $W^{\lambda,0}$ is the interaction-energy integrand in the conventional density-functional adiabatic connection.

The authors note that the monotonic decrease of $W^{\lambda,0}$ with respect to $\lambda$ is known, whereas its convexity remains unresolved. Establishing convexity would provide an important structural property of the adiabatic-connection integrand and strengthen the mathematical foundations of density functional theory.