Given even strongly log-concave random vectors \(X_{0}\) and \(X_{1}\) in \(\mathbb {R}^n\) , we show that a natural joint distribution \((X_{0},X_{1})\) satisfies \(\begin{aligned} e^{ - \frac{1}{n}D ((1-t)X_{0} + t X_{1} \Vert Z)} \ge (1-t) e^{ - \frac{1}{n}D (X_{0} \Vert Z)} + t e^{ - \frac{1}{n}D ( X_{1} \Vert Z)}, \end{aligned}\) where Z is distributed according to the standard Gaussian measure \(\gamma \) on \(\mathbb {R}^n\) , \(t \in [0,1]\) , and \(D(\cdot \Vert Z)\) is the Gaussian relative entropy. This extends and provides a different viewpoint on the corresponding geometric inequality proved by Eskenazis and Moschidis [17], namely that \(\begin{aligned} \gamma \left( (1-t) K_{0} + t K_{1} \right) ^{\frac{1}{n}} \ge (1-t) \gamma (K_{0})^{\frac{1}{n}} + t \gamma (K_{1})^{\frac{1}{n}}, \end{aligned}\) when \(K_{0}, K_{1} \subseteq \mathbb {R}^n\) are origin-symmetric convex bodies. As an application, using Donsker–Varadhan duality, we obtain Gaussian Borell–Brascamp–Lieb inequalities applicable to even log-concave functions, which serve as functional forms of the Eskenazis–Moschidis inequality.
