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<p>Given a normed plane <i>X</i>, a subset <i>K</i> of <i>X</i> is non-centerable if its diameter is smaller than twice its Chebyshev radius. We prove that for any non-centerable set <i>K</i>, the only Chebyshev center is the circumcenter of three points of <i>K</i>, which is an interior point of the convex hull of <i>K</i>. We give a simple and direct proof that the Jung constant of <i>X</i> is 4/3 if and only if the unit sphere is a regular hexagon. We present some examples of normed planes whose Jung constant is equal to <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(2/\sqrt{3}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mo stretchy="false">/</mo> <msqrt> <mn>3</mn> </msqrt> </mrow> </math></EquationSource> </InlineEquation>, as in the Ecuclidean case, in particular those whose unit sphere is invariant under rotations of angle <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\pi /6\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>π</mi> <mo stretchy="false">/</mo> <mn>6</mn> </mrow> </math></EquationSource> </InlineEquation> radians. We compute some estimates of the Jung constant for different <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\ell _p\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>ℓ</mi> <mi>p</mi> </msub> </math></EquationSource> </InlineEquation> normed planes.</p>

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Wheeling around Chebyshev centers and Jung constant in normed planes

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2026年7月9日

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  • Javier Alonso,
  • Pedro Martín,
  • Pier Luigi Papini

摘要

Given a normed plane X, a subset K of X is non-centerable if its diameter is smaller than twice its Chebyshev radius. We prove that for any non-centerable set K, the only Chebyshev center is the circumcenter of three points of K, which is an interior point of the convex hull of K. We give a simple and direct proof that the Jung constant of X is 4/3 if and only if the unit sphere is a regular hexagon. We present some examples of normed planes whose Jung constant is equal to \(2/\sqrt{3}\) 2 / 3 , as in the Ecuclidean case, in particular those whose unit sphere is invariant under rotations of angle \(\pi /6\) π / 6 radians. We compute some estimates of the Jung constant for different \(\ell _p\) ℓ p normed planes.