Abstract
Let X be a non-elementary Riemann surface of type (g, n), where g is the number of genus and n is the number of punctures with 3g-3+n > 1. Let T(X) be the Teichmuller space of X. By constructing a certain subset E of T(X), we show that the convex hull of E with respect to the Teichmuller metric, the Caratheodory metric and the Weil-Petersson metric is not in any thick part of the Teichmuller space, respectively. This implies that convex hulls of thick part of Teichmuller space with respect to these metrics are not always in thick part of Teichmuller space, as well as the facts that thick part of Teichmuller space is not always convex with respect to these metrics.
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