We consider the problem faced by a firm that receives highly differentiated products in an online fashion. The firm needs to price these products to sell them to its customer base. Products are described by vectors of features and the market value of each product is linear in the values of the features. The firm does not initially know the values of the different features, but can learn the values of the features based on whether products were sold at the posted prices in the past. This model is motivated by applications such as online marketplaces, online flash sales, and loan pricing. We first consider a multidimensional version of binary search over polyhedral sets and show that it has a worst-case regret which is exponential in the dimension of the feature space. We then propose a modification of the prior algorithm where uncertainty sets are replaced by their Löwner-John ellipsoids. We show that this algorithm has a worst-case regret which is quadratic in the dimension of the feature space and logarithmic in the time horizon. We also show how to adapt our algorithm to the case where valuations are noisy. Finally, we present computational experiments to illustrate the performance of our algorithm. This paper was accepted by Yinyu Ye, optimization.