A Uniform-Distribution Counterexample to Proposed Log-Concave Concentration Extremal Bounds

We give an elementary counterexample to both equalities proposed in Conjecture 1.3 of Pinelis
concerning one-sided concentration of mean-zero, variance-one, log-concave distributions. The centered,
standardized uniform distribution has smaller density at the mean than the proposed translated exponential
extremizer and smaller one-sided interval mass. We identify the exact interval-length threshold for this
comparison and show that the obstruction persists for smooth, strictly positive log-concave densities. The
counterexample does not challenge the proved universal lower bounds.