A sequence of positive real numbers $a_1, a_2, \ldots, a_n$, is log-concave if $a_i^2 \geq a_{i-1}a_{i+1}$ for all $i$ ranging from 2 to $n-1$. Log-concavity naturally arises in various aspects of mathematics, each characterized by different underlying mechanisms. Examples range from log-concave inequalities that are provable through elementary means, such as the binomial coefficients $a_i = \binom{n}{i}$, to intricate inequalities that requires sophisticated technologies to prove, such as the celebrated Stanley's inequality for linear extensions of posets. It is then natural to ask if the latter type of inequalities is intrinsically more challenging than the former. In this talk, we discuss a rigorous framework to approach this type of questions, by employing a combination of combinatorics, complexity theory, and geometry. This is a joint work with Igor Pak.
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