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Abstract A new constant WD ( X ) $\mathit{WD}(X)$ is introduced into any real 2 n $2^{n}$ -dimensional symmetric normed space X. By virtue of this constant, an upper bound of the geometric constant D ( X ) $D(X)$ , which is used to measure the difference between Birkhoff orthogonality and isosceles orthogonality, is obtained and further extended to an arbitrary m-dimensional symmetric normed linear space ( m ≥ 2 $m\geq2$ ). As an application, the result is used to prove a special case for the reverse Hölder inequality.