Borell's Generalized Prékopa-Leindler Inequality: A Simple Proof

We present a simple proof of Christer Borell's general inequality in the Brunn-Minkowski theory. We then discuss applications of Borell's inequality to the log-Brunn-Minkowski inequality of K. J. Böröczky, E. Lutwak, D. Yang and G. Zhang [The log-Brunn-Minkowski inequality, Adv. Math. 231 (2012) 1974-1997].