<p>In this paper, we study even weighted <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2821_Article_IEq4.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Minkowski problems for weights on <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2821_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="20" /> </InlineMediaObject> <EquationSource Format="TEX">$\mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </math></EquationSource> </InlineEquation> satisfying a new decay condition. As two main results, we prove the existence of solutions for <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2821_Article_IEq6.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$p\ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, as well as the uniqueness for <InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="9_2025_2821_Article_IEq7.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">$p\in \mathbb {R}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>∈</mo> <mi mathvariant="double-struck">R</mi> </mrow> </math></EquationSource> </InlineEquation>. Additionally, we also establish two logarithmic Minkowski type inequalities.</p>

A New Approach to Even Weighted $L_p$ Minkowski Problems

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发表日期：

2025年3月5日

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Mengya Chang,
Hai Li

摘要

In this paper, we study even weighted $L_p$ $L_{p}$ Minkowski problems for weights on $\mathbb {R}^n$ $R^{n}$ satisfying a new decay condition. As two main results, we prove the existence of solutions for $p\ge 1$ $p \geq 1$ , as well as the uniqueness for $p\in \mathbb {R}$ $p \in R$ . Additionally, we also establish two logarithmic Minkowski type inequalities.

A New Approach to Even Weighted \(L_p\) Minkowski Problems

摘要