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We show that a homogeneous Siegel domain is symmetric if and only if its Cayley transform image is convex. Moreover, this convexity forces the pa- rameter of the Cayley transform to be a specific one, so that the Cayley transform coincides with the inverse of the Cayley transform introduced by Koranyi and Wolf.

A characterization of symmetric Siegel domains by convexity of Cayley transform images