Minimum R\'enyi Entropy of a Log-Concave Random Variable at Fixed Variance for Orders Below One

Bia\l{}obrzeski and Nayar stated as open the problem of minimizing the R\'enyi entropy of order $\alpha\in(0,2)$ over log-concave random variables with a given variance. Scaling leaves unchanged the entropy minus half the logarithm of the variance, so we minimize this quantity. Melbourne, Nayar and Roberto later proved that for $\alpha\ge1$ the one-sided exponential law minimizes this quantity. We solve the problem for the remaining orders $\alpha\in(0,1)$. The minimum over all log-concave laws equals the minimum over the family of exponential densities truncated to an interval, which has the uniform law at rate zero and the one-sided exponential law in the limit of infinite rate. For $\alpha\le3/5$, the uniform law attains the minimum, with the value $\frac12\log12$. For $\alpha\in(3/5,1)$, a density of finite positive rate attains it, and the minimum is below the values of the quantity at those two laws. The proof adapts a published reduction to non-increasing densities whose logarithm has two affine pieces. We write the level-set widths of such a density as an integral against a measure on the levels, which makes the integrals of the density and of its power $\alpha$ linear in that measure. The derivative in that measure gives necessary conditions at a minimizer. Interval arithmetic, which encloses the values of a function on ranges of its arguments, shows that each two-piece density violates a condition or has a value of the quantity above the minimum over the truncated family, apart from two parameter ranges that lemmas cover. The computation's scripts are supplementary material.