On the functional Minkowski problem

Abstract To every log-concave function f one may associate a pair of measures $$(\mu _{f},\nu _{f})$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> which are the surface area measures of f . These are a functional extension of the classical surface area measure of a convex body, and measure how the integral $$\int f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>∫</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> changes under perturbations. The functional Minkowski problem then asks which pairs of measures can be obtained as the surface area measures of a log-concave function. In this work, we fully solve this problem. Furthermore, we prove that the surface area measures are continuous with respect to a suitable topology: If $$f_{k}\rightarrow f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> , then $$\left( \mu _{f_{k}},\nu _{f_{k}}\right) \rightarrow \left( \mu _{f},\nu _{f}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:msub> <mml:mi>μ</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> </mml:mfenced> <mml:mo>→</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> in a corresponding sense. Finding the appropriate mode of convergence of the pairs $$\left( \mu _{f_{k}},\nu _{f_{k}}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfenced> <mml:msub> <mml:mi>μ</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> </mml:mfenced> </mml:math> sheds a new light on the construction of functional surface area measures. To prove this continuity theorem we associate to every convex function a new type of radial function, which seems to be an interesting construction on its own right. Finally, we prove that the solution to the functional Minkowski problem is continuous in the data, in the sense that if $$\left( \mu _{f_{k}},\nu _{f_{k}}\right) \rightarrow \left( \mu _{f},\nu _{f}\right) $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfenced> <mml:msub> <mml:mi>μ</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> </mml:msub> </mml:mfenced> <mml:mo>→</mml:mo> <mml:mfenced> <mml:msub> <mml:mi>μ</mml:mi> <mml:mi>f</mml:mi> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>ν</mml:mi> <mml:mi>f</mml:mi> </mml:msub> </mml:mfenced> </mml:mrow> </mml:math> then $$f_{k}\rightarrow f$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>f</mml:mi> <mml:mi>k</mml:mi> </mml:msub> <mml:mo>→</mml:mo> <mml:mi>f</mml:mi> </mml:mrow> </mml:math> up to translations.