Let M be a convex body and let K be a closed convex surface K both contained in the Euclidean space <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mathbb {E}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation>. What can we say about M if K encloses M and if from all the points in K the body M looks the same? In this work we are going to present a result which claims that if for every two support cones <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$C_x$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>x</mi> </msub> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$C_y$</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>C</mi> <mi>y</mi> </msub> </math></EquationSource> </InlineEquation> of M, with apexes <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$x,y \in K$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>∈</mo> <mi>K</mi> </mrow> </math></EquationSource> </InlineEquation>, respectively, there exists <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\Phi $</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Φ</mi> </math></EquationSource> </InlineEquation> in the semi direct product of the orthogonal group O(3) and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\mathbb {E}^3$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mi mathvariant="double-struck">E</mi> </mrow> <mn>3</mn> </msup> </math></EquationSource> </InlineEquation> such that <Equation ID="Equ18"> <EquationSource Format="TEX">$C_y=\Phi (C_x),$</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <msub> <mi>C</mi> <mi>y</mi> </msub> <mo>=</mo> <mi mathvariant="normal">Φ</mi> <mrow> <mo stretchy="false">(</mo> <msub> <mi>C</mi> <mi>x</mi> </msub> <mo stretchy="false">)</mo> </mrow> <mo>,</mo> </mrow> </math></EquationSource> </Equation>and this can be done in a continuous way, then M is a sphere.

Characterization of the sphere by means of congruent support cones

期刊论文
发表日期：

2025年10月13日

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E. Morales Amaya

摘要

Let M be a convex body and let K be a closed convex surface K both contained in the Euclidean space $\mathbb {E}^3$ $E^{3}$ . What can we say about M if K encloses M and if from all the points in K the body M looks the same? In this work we are going to present a result which claims that if for every two support cones $C_x$ $C_{x}$ , $C_y$ $C_{y}$ of M, with apexes $x,y \in K$ $x, y \in K$ , respectively, there exists $\Phi $ $Φ$ in the semi direct product of the orthogonal group O(3) and $\mathbb {E}^3$ $E^{3}$ such that $C_y=\Phi (C_x),$ $C_{y} = Φ (C_{x}),$ and this can be done in a continuous way, then M is a sphere.