the probability distribution is concentrated in a contiguous interval of small measure. To formalize this distinction, we say that two probability density functions and are equimeasurable if where is the Lebesgue measure. Any two equimeasurable probability density functions have the same Shannon entropy, and in fact the same Rényi entropy, of any order. The same is not true of variance, however. Any probability density function has a radially decreasing equimeasurable "rearrangement" whose variance is less (up to translation) than any other rearrangement of the function; and there exist rearrangements of arbitrarily high variance, (all having the same entropy.) See also Inequalities in information theory Logarithmic Schrödinger equation Uncertainty principle Riesz–Thorin theorem Fourier transform References Further reading Jizba, P.; Ma,Y.; Hayes, A.; Dunningham, J.A. (2016). "One-parameter class of uncertainty relations based on entropy power". Phys. Rev. E 93 (6): 060104(R). doi:10.1103/PhysRevE.93.060104. arXiv:math/0605510v1 Quantum mechanical entropy Information theory Physical quantities Inequalities