<p>The (unbounded version of the) Lempert function <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$l_D$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>l</mi> <mi>D</mi> </msub> </math></EquationSource> </InlineEquation> on a domain <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$D\subset \mathbb {C}^d$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>D</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> does not usually satisfy the triangle inequality, but on bounded <InlineEquation ID="IEq3"> <EquationSource Format="TEX">${\mathcal {C}}^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="script">C</mi> </mrow> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation>-smooth strictly pseudoconvex domains, it satisfies a quasi-triangle inequality: <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$l_D(a,c)\le C( l_D(a,b)+l_D(b,c))$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>l</mi> <mi>D</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo>≤</mo> <mi>C</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>l</mi> <mi>D</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>a</mi> <mo>,</mo> <mi>b</mi> <mo stretchy="false">)</mo> </mrow> <mo>+</mo> <msub> <mi>l</mi> <mi>D</mi> </msub> <mrow> <mo stretchy="false">(</mo> <mi>b</mi> <mo>,</mo> <mi>c</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. We show that pseudoconvexity is necessary for this property as soon as <i>D</i> has a <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathcal C^1$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi mathvariant="script">C</mi> <mn>1</mn> </msup> </math></EquationSource> </InlineEquation>-smooth boundary. We also give estimates of the Lempert function and of other invariants in some domains which are models for local situations, and derive some general local bounds depending on the regularity of the boundary of a domain.</p>

Quasi-triangle inequality for the Lempert function

期刊论文
发表日期：

2025年11月25日

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Nikolai Nikolov,
Pascal J. Thomas

摘要

The (unbounded version of the) Lempert function $l_D$ $l_{D}$ on a domain $D\subset \mathbb {C}^d$ $D \subset C^{d}$ does not usually satisfy the triangle inequality, but on bounded ${\mathcal {C}}^2$ $C^{2}$ -smooth strictly pseudoconvex domains, it satisfies a quasi-triangle inequality: $l_D(a,c)\le C( l_D(a,b)+l_D(b,c))$ $l_{D} (a, c) \leq C (l_{D} (a, b) + l_{D} (b, c))$ . We show that pseudoconvexity is necessary for this property as soon as D has a $\mathcal C^1$ $C^{1}$ -smooth boundary. We also give estimates of the Lempert function and of other invariants in some domains which are models for local situations, and derive some general local bounds depending on the regularity of the boundary of a domain.