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A new upper bound of geometric constant [Formula: see text].
A new constant [Formula: see text] is introduced into any real [Formula: see text]-dimensional symmetric normed space X. By virtue of this constant, an upper bound of the geometric constant [Formula: see text], 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 ([Formula: see text]). As an application, the result is used to prove a special case for the reverse Hölder inequality.
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