| Summary: | In this paper, we firstly investigate the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> proposed by Gao further by discussing several properties of it that have not yet been discovered. Secondly, we focus on a new constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> closely related to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>H</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula>, along with a variety of geometric properties. In addition, we show several relations among it and the several basic geometric constants via a few inequalities. Finally, we manage to characterize the geometric properties of its generalized forms <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>G</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> explicitly.
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