A celebrated result of Black (1948a) demonstrates the existence of a simple-majority winner when preferences are single-peaked. The social choice follows the preferences of the median voter: the median voter's most-preferred outcome beats any alternative. However, this conclusion does not extend to elections in which candidates differ in more than one dimension. This paper provides a multi-dimensional analog of the median voter result. We provide conditions under which the mean voter's most preferred outcome is unbeatable according to a 64%-majority rule. The conditions supporting this result represent a significant generalization of Caplin and Nalebuff (1988). The proof of our mean voter result uses a mathematical aggregation theorem due to Prékopa (1971, 1973) and Borell (1975). This theorem has broad applications in economics. An application to the distribution of income is described at the end of this paper; results on imperfect competition are presented in the companion paper, Caplin and Nalebuff (1991).