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In most variants of the Hotelling-Downs model of election, it is assumed that voters have concave utility functions. This assumption is arguably justied in issues such as economic policies, but convex utilities are perhaps more appropriate in others such as moral or religious issues. In this paper we analyze the implications of convex utility functions in a two-candidate probabilistic voting model with a polarized voter distribution. We show that the equilibrium policies diverge if and only if voters' utility function is suciently convex. If two or more issues are involved, policies converge in "concave issues" and diverge in "convex issues."