<p>The <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Monge-Ampère equation that was devoted by Cheng-Yau for the classical case <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$p=1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo>=</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and by Chou-Wang for general <i>p</i> is considered. In particular, it is proved that as <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$p\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, the solution to the <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Monge-Ampère equation converges uniformly to the constant 1. In general, it is proved that as <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$p\rightarrow \infty $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>p</mi> <mo stretchy="false">→</mo> <mi>∞</mi> </mrow> </math></EquationSource> </InlineEquation>, the solution to the <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$L_p$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>L</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> Minkowski problem initiated by Lutwak converges in the Hausdorff metric to the Wulff shape of the data measure. If in addition the support of the data measure is the unit sphere, the ultimate shape is the unit ball.</p>

The ultimate shape of solution to the $\varvec{L_p}$ Minkowski problem

期刊论文
发表日期：

2026年3月7日

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Du Zou

摘要

The $L_p$ $L_{p}$ Monge-Ampère equation that was devoted by Cheng-Yau for the classical case $p=1$ $p = 1$ and by Chou-Wang for general p is considered. In particular, it is proved that as $p\rightarrow \infty $ $p \to \infty$ , the solution to the $L_p$ $L_{p}$ Monge-Ampère equation converges uniformly to the constant 1. In general, it is proved that as $p\rightarrow \infty $ $p \to \infty$ , the solution to the $L_p$ $L_{p}$ Minkowski problem initiated by Lutwak converges in the Hausdorff metric to the Wulff shape of the data measure. If in addition the support of the data measure is the unit sphere, the ultimate shape is the unit ball.

The ultimate shape of solution to the \(\varvec{L_p}\) Minkowski problem

摘要