The<i>L</i><sub><i>p</i></sub>chord Minkowski problem
Dongmeng Xi, Deane Yang, Gaoyong Zhang, Yiming Zhao
Abstract
Abstract Chord measures are newly discovered translation-invariant geometric measures of convex bodies in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msup><m:mrow><m:mi mathvariant="double-struck">R</m:mi></m:mrow><m:mrow><m:mi>n</m:mi></m:mrow></m:msup></m:math> {{\mathbb{R}}}^{n} , in addition to Aleksandrov-Fenchel-Jessen’s area measures. They are constructed from chord integrals of convex bodies and random lines. Prescribing the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msub></m:math> {L}_{p} chord measures is called the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msub></m:math> {L}_{p} chord Minkowski problem in the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msub></m:math> {L}_{p} Brunn-Minkowski theory, which includes the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msub></m:math> {L}_{p} Minkowski problem as a special case. This article solves the <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:msub><m:mrow><m:mi>L</m:mi></m:mrow><m:mrow><m:mi>p</m:mi></m:mrow></m:msub></m:math> {L}_{p} chord Minkowski problem when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>p</m:mi><m:mo>></m:mo><m:mn>1</m:mn></m:math> p\gt 1 and the symmetric case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mn>0</m:mn><m:mo><</m:mo><m:mi>p</m:mi><m:mo><</m:mo><m:mn>1</m:mn></m:math> 0\lt p\lt 1 .