For subdifferentiable convex functions, strict convexity holds exactly when the subdifferential is strictly monotone, with extensions of related characterizations to Hilbert spaces.
IUSEM, On some properties of paramonotone operators,Journal of Convex Analysis5 (1998), 269–278
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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.