Let <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1030_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="93" /> </InlineMediaObject> <EquationSource Format="TEX">$ f: \mathbb {R}^n \rightarrow \mathbb {R}^m $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>f</mi> <mo>:</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo stretchy="false">→</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>m</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> be a <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1030_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="21" /> </InlineMediaObject> <EquationSource Format="TEX">$C^2$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mi>C</mi> <mn>2</mn> </msup> </math></EquationSource> </InlineEquation> definable map in an o-minimal structure. We prove that the Lipschitz-Killing curvature density at infinity <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1030_Article_IEq3.gif" Format="GIF" Height="21" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">$\Lambda _k^{\lim }(f^{-1}(t), \infty )$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msubsup> <mi mathvariant="normal">Λ</mi> <mi>k</mi> <mo movablelimits="true">lim</mo> </msubsup> <mrow> <mo stretchy="false">(</mo> <msup> <mi>f</mi> <mrow> <mo>-</mo> <mn>1</mn> </mrow> </msup> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mi>∞</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> of the fibers is locally Lipschitz outside the set of asymptotic critical values of f for <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1030_Article_IEq4.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">$k \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>. For <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="29_2025_1030_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">$k = 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>, it is locally Lipschitz outside the set of generalized critical values of f. This reinforces the recent result of Dutertre and Grandjean, where only continuity was achieved.

Lipschitz continuity of Lipschitz-Killing curvature densities at infinity

期刊论文
发表日期：

2025年3月17日

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Si Tiep Dinh,
Nhan Nguyen

摘要

Let $ f: \mathbb {R}^n \rightarrow \mathbb {R}^m $ $f : R^{n} \to R^{m}$ be a $C^2$ $C^{2}$ definable map in an o-minimal structure. We prove that the Lipschitz-Killing curvature density at infinity $\Lambda _k^{\lim }(f^{-1}(t), \infty )$ $Λ_{k}^{lim} (f^{- 1} (t), \infty)$ of the fibers is locally Lipschitz outside the set of asymptotic critical values of f for $k \ge 1$ $k \geq 1$ . For $k = 0$ $k = 0$ , it is locally Lipschitz outside the set of generalized critical values of f. This reinforces the recent result of Dutertre and Grandjean, where only continuity was achieved.