For subdifferentiable convex functions, strict convexity holds exactly when the subdifferential is strictly monotone, with extensions of related characterizations to Hilbert spaces.
BECK,First-Order Methods in Optimization, SIAM
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On Characterizations of (Almost) Strictly Convex Functions
For subdifferentiable convex functions, strict convexity holds exactly when the subdifferential is strictly monotone, with extensions of related characterizations to Hilbert spaces.