Monotonicity of the Gaussian measure under Banaszczyk transforms

In the proof of his famous 5K-theorem, W. Banaszczyk introduced a transformation of convex bodies for which the Gaussian measure is monotone. In this note, we present a simplified proof of this monotonicity by slightly modifying Banaszczyk's transform, so that it interacts smoothly with Ehrhard symmetrizations, thereby yielding a somewhat easier proof of the 5K-theorem.