<p>We consider when a subset <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2024_1500_Article_IEq1.gif" Format="GIF" Height="17" Rendition="HTML" Resolution="72" Type="Linedraw" Width="59" /> </InlineMediaObject> <EquationSource Format="TEX">$X\subset\mathbb{R}^{d}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>d</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> has a Radon partition <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2024_1500_Article_IEq2.gif" Format="GIF" Height="16" Rendition="HTML" Resolution="72" Type="Linedraw" Width="98" /> </InlineMediaObject> <EquationSource Format="TEX">$X=X_{1}\sqcup X_{2}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>X</mi> <mo>=</mo> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo>⊔</mo> <msub> <mi>X</mi> <mn>2</mn> </msub> </mrow> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ1"> <MediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2024_1500_Article_Equ1.gif" Format="GIF" Height="19" Rendition="HTML" Resolution="72" Type="Linedraw" Width="448" /> </MediaObject> <EquationSource Format="TEX">$\dim(({\rm conv} X_{1})\cap({\rm conv} X_{2}) )= \min\lbrace \dim({\rm conv} X_{1}), \dim({\rm conv} X_{2})\rbrace,$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mo>dim</mo> <mrow> <mo stretchy="false">(</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">conv</mi> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>∩</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">conv</mi> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mo movablelimits="true">min</mo> <mrow> <mo stretchy="false">{</mo> <mo>dim</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">conv</mi> <msub> <mi>X</mi> <mn>1</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> <mo>dim</mo> <mrow> <mo stretchy="false">(</mo> <mi mathvariant="normal">conv</mi> <msub> <mi>X</mi> <mn>2</mn> </msub> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">}</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation> showing that such a partition always exists when <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2024_1500_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">$X$</EquationSource> <EquationSource Format="MATHML"><math> <mi>X</mi> </math></EquationSource> </InlineEquation> has at least <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10474_2024_1500_Article_IEq4.gif" Format="GIF" Height="23" Rendition="HTML" Resolution="72" Type="Linedraw" Width="60" /> </InlineMediaObject> <EquationSource Format="TEX">$\lfloor\frac{3d}{2}\rfloor+2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo>⌊</mo> <mfrac> <mrow> <mn>3</mn> <mi>d</mi> </mrow> <mn>2</mn> </mfrac> <mo>⌋</mo> <mo>+</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> points in general position. The latter bound is sharp.</p>

Dimension of the Radon set

期刊论文
发表日期：

2025年2月24日

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S. B. Choudhury,
S. Deo,
D. Gauld,
S. Podder

摘要

We consider when a subset $X\subset\mathbb{R}^{d}$ $X \subset R^{d}$ has a Radon partition $X=X_{1}\sqcup X_{2}$ $X = X_{1} ⊔ X_{2}$ such that $\dim(({\rm conv} X_{1})\cap({\rm conv} X_{2}) )= \min\lbrace \dim({\rm conv} X_{1}), \dim({\rm conv} X_{2})\rbrace,$ $dim ((conv X_{1}) \cap (conv X_{2})) = min {dim (conv X_{1}), dim (conv X_{2})},$ showing that such a partition always exists when $X$ $X$ has at least $\lfloor\frac{3d}{2}\rfloor+2$ $⌊ \frac{3 d}{2} ⌋ + 2$ points in general position. The latter bound is sharp.