When Does the Equality J(X~*) = J(X) Hold for a Two-dimensional Banach Space X ?

In this paper, we consider the following problem about the James constant: When does the equality J(X*) = J(X) hold for a Banach space X ? It is known that the James constant of a Banach space does not coincide with that of its dual space in general. In fact, we already have counterexamples of two-dimensional normed spaces that are equipped with either symmetric or absolute norms. However,we show that if the norm on a two-dimensional space X is both symmetric and absolute, then the equality J(X*) = J(X) holds. This provides a global answer to the problem in the two-dimensional case.