For subdifferentiable convex functions, strict convexity holds exactly when the subdifferential is strictly monotone, with extensions of related characterizations to Hilbert spaces.
STROMBERG,An Introduction to Classical Real Analysis, AMS Chelsea Publishing, Providence, RI
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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.