<p>For a metric space (<i>X</i>,&#xa0;<i>d</i>), a subset <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(A \subset X\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mo>⊂</mo> <mi>X</mi> </mrow> </math></EquationSource> </InlineEquation> is said to resolve <i>X</i> if every point in <i>X</i> is uniquely determined by its distances to the points in <i>A</i>. The metric dimension of <i>X</i> is defined as the minimal cardinality among all subsets of <i>X</i> that resolve <i>X</i>. In this paper, we prove that the metric dimensions of the complex hyperbolic space <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {H}_{\mathbb {C}}^n\)</EquationSource> <EquationSource Format="MATHML"><math> <msubsup> <mi mathvariant="double-struck">H</mi> <mrow> <mi mathvariant="double-struck">C</mi> </mrow> <mi>n</mi> </msubsup> </math></EquationSource> </InlineEquation> and the Heisenberg group <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathcal {H}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">H</mi> </math></EquationSource> </InlineEquation> are <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(2n+1\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>2</mn> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> and 4, respectively.</p>

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The Metric Dimensions of the Complex Hyperbolic Space and the Heisenberg Group

  • Fangming Cai,
  • Wei Liao

摘要

For a metric space (Xd), a subset \(A \subset X\) A X is said to resolve X if every point in X is uniquely determined by its distances to the points in A. The metric dimension of X is defined as the minimal cardinality among all subsets of X that resolve X. In this paper, we prove that the metric dimensions of the complex hyperbolic space \(\mathbb {H}_{\mathbb {C}}^n\) H C n and the Heisenberg group \(\mathcal {H}\) H are \(2n+1\) 2 n + 1 and 4, respectively.