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Orthogonal frequency division multiplexing (OFDM) is an attractive technique for high data rate transmissions in wireless and wireline systems. However OFDM signals suffer from high peak-to-mean envelope power (PMEPR). In this paper we consider the problem of reducing this high PMEPR using reserved peak reduction subcarriers as considered by Tellado and Cioffi. We analyze the fundamental tradeoff between PMEPR reduction and the rate loss due to reserved subcarriers. We first provide a lower bound on the complementary cumulative distribution of PMEPR using infinitely many reserved subcarriers. We then show that the problem of minimizing the maximum of the absolute value of the signal using reserved subcarriers can be stated as a convex linear matrix inequality problem. While the problem is proved to be convex, its complexity is cubic in the number of OFDM subcarriers n which can be prohibitive when n is large. We then propose a suboptimal greedy algorithm based on p-norm minimization that chooses only bipolar values for each subcarrier that has less complexity and more PMEPR reduction than the previously proposed algorithm in. Our results provide the best tradeoff possible between PMEPR reduction and the associated rate loss and also leads to a practical algorithm to approach those limits.