| Summary: | We will introduce some new geometric constants based on 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 and the constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>A</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> proposed by M. Baronti et al. We first provide a study of a new constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> closely related to the midlines of equilateral triangles, including a discussion of some of its properties and the connections with other parameters of the sphere. Next, we focus on a new constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and its generalized form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, along with some of their basic properties. Finally, we concentrate on a new constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mn>3</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and discuss some of its properties.
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