The Wills functional [Formula: see text] of a convex body [Formula: see text], defined as the sum of its intrinsic volumes [Formula: see text], turns out to have many interesting applications and properties. In this paper, we make profit of the fact that it can be represented as the integral of a log-concave function, which, furthermore, is the Asplund product of other two log-concave functions, and obtain new properties of the Wills functional (indeed, we will work in a more general setting). Among others, we get upper bounds for [Formula: see text] in terms of the volume of [Formula: see text], as well as Brunn–Minkowski and Rogers–Shephard-type inequalities for this functional. We also show that the cube of edge-length 2 maximizes [Formula: see text] among all [Formula: see text]-symmetric convex bodies in John position, and we reprove the well-known McMullen’s inequality [Formula: see text] using a different approach.

中文翻译：

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（广义）遗嘱泛函的进一步不等式

凸体 [公式： 见文本] 的 Wills 泛函 [公式： 见文本]，定义为其内在体积 [公式： 见文本] 的总和，结果证明有许多有趣的应用和属性。在本文中，我们利用它可以表示为对数凹函数的积分这一事实，并且它是其他两个对数凹函数的 Asplund 乘积，并获得了 Wills 泛函的新性质 (事实上，我们将在更一般的环境中工作）。除其他外，我们根据 [Formula: see text] 的体积得到 [Formula: see text] 的上限，以及该泛函的 Brunn-Minkowski 和 Rogers-Shephard 型不等式。我们还证明了边长为 2 的立方体在约翰位置的所有 [公式：见文本] 对称凸体中最大化 [公式：见文本]，