A short, based on the mixed volume, proof of Liggett's theorem on the   convolution of ultra-logconcave sequences
  R. Pemantle conjectured, and T.M. Liggett proved in 1997, that the convolution of two ultra-logconcave is ultra-logconcave. Liggett's proof is elementary but long. We present here a short proof, based on the mixed volume of convex sets. 
Leonid Gurvits