A new constant mathit{WD}(X) is introduced into any real 2^{n}-dimensional symmetric normed space X. By virtue of this constant, an upper bound of the geometric constant 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 (mgeq2). As an application, the result is used to prove a special case for the reverse Hölder inequality.