<p>In this work we prove that if for a pair of convex bodies <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$K_1, K_2 \subset \mathbb {R}^n$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>K</mi> <mn>1</mn> </msub> <mo>,</mo> <msub> <mi>K</mi> <mn>2</mn> </msub> <mo>⊂</mo> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n \ge 3$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>≥</mo> <mn>3</mn> </mrow> </math></EquationSource> </InlineEquation>, there exists a hyperplane <i>H</i> and two distinct points <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$p_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$p_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>p</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\mathbb {R}^n \setminus H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msup> <mrow> <mi mathvariant="double-struck">R</mi> </mrow> <mi>n</mi> </msup> <mo lspace="0.15em" rspace="0.15em" stretchy="false">\</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> such that for every <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$(n-2)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>-</mo> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation>-plane <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$M \subset H$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>M</mi> <mo>⊂</mo> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>, there exists a reflection mapping the hypersection of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$K_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq9"> <EquationSource Format="TEX">$\textrm{aff}\{p_1, M\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>aff</mtext> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mn>1</mn> </msub> <mo>,</mo> <mi>M</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation> onto the hypersection of <InlineEquation ID="IEq10"> <EquationSource Format="TEX">$K_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation> defined by <InlineEquation ID="IEq11"> <EquationSource Format="TEX">$\textrm{aff}\{p_2, M\}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mtext>aff</mtext> <mo stretchy="false">{</mo> <msub> <mi>p</mi> <mn>2</mn> </msub> <mo>,</mo> <mi>M</mi> <mo stretchy="false">}</mo> </mrow> </math></EquationSource> </InlineEquation>, then there exists a reflection which maps <InlineEquation ID="IEq12"> <EquationSource Format="TEX">$K_1$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> onto <InlineEquation ID="IEq13"> <EquationSource Format="TEX">$K_2$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>K</mi> <mn>2</mn> </msub> </math></EquationSource> </InlineEquation>.</p>

Convex bodies with pairs of sections associated by reflections

期刊论文
发表日期：

2025年9月2日

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E. Morales-Amaya

摘要

In this work we prove that if for a pair of convex bodies $K_1, K_2 \subset \mathbb {R}^n$ $K_{1}, K_{2} \subset R^{n}$ , $n \ge 3$ $n \geq 3$ , there exists a hyperplane H and two distinct points $p_1$ $p_{1}$ and $p_2$ $p_{2}$ in $\mathbb {R}^n \setminus H$ $R^{n} \ H$ such that for every $(n-2)$ $(n - 2)$ -plane $M \subset H$ $M \subset H$ , there exists a reflection mapping the hypersection of $K_1$ $K_{1}$ defined by $\textrm{aff}\{p_1, M\}$ $aff {p_{1}, M}$ onto the hypersection of $K_2$ $K_{2}$ defined by $\textrm{aff}\{p_2, M\}$ $aff {p_{2}, M}$ , then there exists a reflection which maps $K_1$ $K_{1}$ onto $K_2$ $K_{2}$ .