A lower bound on the Rényi differential entropy of a sum of independent random vectors is demonstrated in terms of rearrangements. For the special case of Boltzmann-Shannon entropy, this lower bound is better than that given by the entropy power inequality. Several applications are discussed, including a new proof of the classical entropy power inequality and an entropy inequality involving symmetrization of Lévy processes.

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman (Ann. Probab., 39(4):1528-1543.Mathematics Subject Classification (2010). Primary 52A40; Secondary 60E15, 94A17.

Excess reactive oxygen species (ROS) have been proved to damage cancer cells efficiently. ROS overproduction is thus greatly desirable for cancer therapy. To date, ROS production is generally uncontrollable and outside cells, which always bring severe side-effects in the vasculature. Since most ROS share a very short half-life and primarily react close to their site of formation, it would be more efficient if excess ROS are controllably produced inside cancer cells. Herein, we report an efficient lysosome-controlled ROS overproduction via a pH-responsive catalytic nanosystem (FeOx-MSNs), which catalyze the decomposition of H2O2 to produce considerable ROS selectively inside the acidic lysosomes (pH 5.0) of cancer cells. After a further incorporation of ROS-sensitive TMB into the nanosystem (FeOx-MSNs-TMB), both a distinct cell labeling and an efficient death of breast carcinoma cells are obtained. This lysosome-controlled efficient ROS overproduction suggests promising applications in cancer treatments.

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