Strong Log-Concavity Does Not Imply Log-Submodularity

We disprove a recent conjecture regarding discrete distributions and their generating polynomials stating that strong log-concavity implies log-submodularity.

[1]  Leonid Gurvits,et al.  On multivariate Newton-like inequalities , 2008, 0812.3687.

[2]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[3]  Nima Anari,et al.  Log-concave polynomials II: high-dimensional walks and an FPRAS for counting bases of a matroid , 2018, STOC.

[4]  Suvrit Sra,et al.  Flexible Modeling of Diversity with Strongly Log-Concave Distributions , 2019, NeurIPS.

[5]  June Huh,et al.  Lorentzian polynomials , 2019, Annals of Mathematics.