Jensen Inequalities Based on Medians, Generalized Centers, and Geometric Medians

Abstract

We study Jensen-type inequalities in which the mean is replaced by certain
robust centers, notably medians and geometric medians. We first recall limitations of a
naive “median Jensen inequality”, and show by counterexample that one cannot simply
replace the mean by the median in the classical Jensen inequality. We then prove a valid
median-based Jensen inequality under a natural symmetry condition, and introduce variants
involving generalized Lp centers and geometric medians in Rd. Finally, we present a refined
Jensen inequality for twice differentiable functions with bounded curvature; this yields a
second-order correction involving the variance of the points and strictly sharpens the classical
Jensen inequality. We discuss applications in statistics, optimization, and machine learning,
with an emphasis on robustness.