Are polynomials with only real zeros log concave functions?

jorgejasso85xvx

jorgejasso85xvx

Answered question

2022-11-15

Are polynomials with only real zeros log concave functions?

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erlent00s

erlent00s

Beginner2022-11-16Added 15 answers

Note, this only works when p has negative roots. Then the coefficients of p are positive. We know by Newton's Inequalities that these coefficients form an ultra log-concave sequence. Hence, by Shephard's Theorem (1960), we can realize p as the relative Steiner polynomial Vol ( K + x L ) = k = 0 n ( n k ) W k ( K ; L ) x k of two convex bodies K , L. From there, the Brunn-Minkowski Inequality gives you log-concavity of p as a univariate function of x.

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