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In this paper, we will make some further discussions on the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</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>Y</mi><mi>J</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce two new geometric constants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Y</mi><mi>J</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and some geometric properties of Banach spaces, including uniformly non-square and uniformly convex. It is worth noting that we point out that uniform non-square spaces can be characterized by the side lengths of some special inscribed triangles of unit ball. Secondly, we establish some inequalities for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>L</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>Y</mi><mi>J</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> and some significant geometric constants, including the James constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>(</mo><mi>X</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the von Neumann-Jordan constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>C</mi><mi>NJ</mi></msub><mrow><mo>(</mo><mi>X</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Finally, we introduce the two new geometric constants <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, and calculate the bounds of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><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>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> as well as the values of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><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>L</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>X</mi><mo>,</mo><mo>▵</mo><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for two Banach spaces.