In a 2019 paper, Barnard and Steinerberger show that for f ∈ L 1 (R), the following autocorrelation inequality holds:L 1 , where the constant 0.411 cannot be replaced by 0.37. In addition to being interesting and important in their own right, inequalities such as these have applications in additive combinatorics where some problems, such as those of minimal difference basis, can be encapsulated by a convolution inequality similar to the above integral. Barnard and Steinerberger suggest that future research may focus on the existence of functions extremizing the above inequality (which is itself related to Brascamp-Lieb type inequalities).We show that for f to be extremal under the above, we must have Date: January 9, 2020. 2010 Mathematics Subject Classification. 26D10 (primary), 39B62 (secondary).