  Considering symmetric strictly convex potentials, a local relationship is
inferred from the virial theorem, based on which a real log-concave function
can be constructed. Using this as a weight function and in such a way that the
virial theorem can still be verified, parameter-free ans\"atze for the
eigenfunctions of the associated Schr\"odinger equation are built. To
illustrate the process, the technique is successfully tested against the
harmonic oscillator, in which it leads to the exact eigenfunctions, and against
the quartic anharmonic oscillator, which is considered the paradigmatic testing
ground for new approaches to the Schr\"odinger equation.
