We analyze the smoothness of the ground state energy of a one-parameter Hamiltonian by studying the differential geometry of the numerical range and continuity of the maximum-entropy inference. The domain of the inference map is the numerical range, a convex compact set in the plane. We show that its boundary, viewed as a manifold, has the same order of differentiability as the ground state energy. We prove that discontinuities of the inference map correspond to C1-smooth crossings of the ground state energy with a higher energy level. Discontinuities may appear only at C1-smooth points of the boundary of the numerical range. Discontinuities exist at all C2-smooth non-analytic boundary points and are essentially stronger than at analytic points or at points which are merely C1-smooth (non-exposed points).
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