  Sequences of experimental ground-state energies for both odd and even $A$ are
mapped onto concave patterns cured from convexities due to pairing and/or shell
effects. The same patterns, completed by a list of excitation energies, give
numerical estimates of the grand potential $\Omega(\beta,\mu)$ for a mixture of
nuclei at low or moderate temperatures $T=\beta^{-1}$ and at many chemical
potentials $\mu.$ The average nucleon number $<{\bf A} >(\beta,\mu)$ then
becomes a continuous variable, allowing extrapolations towards nuclear masses
closer to drip lines. We study the possible concavity of several
thermodynamical functions, such as the free energy and the average energy, as
functions of $<{\bf A} >.$ Concavity, which always occur for the free energy
and is usually present for the average energy, allows easy interpolations and
extrapolations providing upper and lower bounds, respectively, to binding
energies. Such bounds define an error bar for the prediction of binding
energies. Finally we show how concavity and universality are related in the
theory of the nuclear density functional.
