| Summary: | We introduce a new geometric constant <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> based on a generalization of the parallelogram law, which is symmetric and related to the length of the inscribed quadrilateral side of the unit ball. We first investigate some basic properties of this new coefficient. Next, it is shown that, for a Banach space, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mi>i</mi><mi>n</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> becomes 16 if and only if the norm is induced by an inner product. Moreover, its properties and some relations between other well-known geometric constants are studied. Finally, a sufficient condition which implies normal structure is presented.
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