Abstract
The first variation of the total mass of log-concave functions was studied by Colesanti and Fragalà, which includes the Lp mixed volume of convex bodies. Using Colesanti and Fragalà’s first variation formula, we define the geominimal surface area for log-concave functions, and its related affine isoperimetric inequality is also established.
Highlights
As we have known, Minkowski addition is the cornerstone in the classical BrunnMinkowski theory
Since δJð f, gÞ includes the Lp mixed volume, we extend the Lp geominimal surface area to the functional version as follows
In Lemma 5, we prove that the above definition includes the Lp geominimal surface area (4) when p ≥ 1 and restricted f, g to the subclass of log-concave functions
Summary
Minkowski addition (the vector addition of convex bodies) is the cornerstone in the classical BrunnMinkowski theory. Colesanti and Fragalà’s work inspired us a natural way to extend the Lp geominimal surface area for convex bodies to the class of log-concave functions. Since δJð f , gÞ includes the Lp mixed volume, we extend the Lp geominimal surface area to the functional version as follows. In Lemma 5, we prove that the above definition includes the Lp geominimal surface area (4) when p ≥ 1 and restricted f , g to the subclass of log-concave functions. Similar to the geometric case, the unique log-concave function f is called p -Petty functions of f and denoted by Tpf. Using p-Petty functions, we obtain the following analytic inequality with equality conditions involving Gðp1Þðf Þ.
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