  I show that a nonnegative Wigner function that represents a mixture of Fock
states is majorized by the Wigner function of the vacuum state. As a
consequence, the integration of any concave function over the Wigner phase
space has a lower value for the vacuum state than for a mixture of Fock states.
The Shannon differential entropy is an example of such concave function of
significant physical importance. I demonstrate that the very cause of the
majorization lies in the fact that a Wigner function is the result of a
convolution. My proof is based on a new majorization result dedicated to the
convolution of the negative exponential distribution with a precisely
constrained function. I present a geometrical interpretation of the new
majorization property in a discrete setting and extend this relation to a
continuous setting. Findings presented in this article might be expanded upon
to explain why the Wigner function of the vacuum majorizes - beyond mixtures of
Fock states - many other physical states represented by a nonnegative Wigner
function.
