Mining high-utility itemsets (HUIs) is a key data mining task. It consists of discovering groups of items that yield a high profit in transaction databases. A major drawback of traditional high-utility itemset mining algorithms is that they can return a large number of HUIs. Analyzing a large result set can be very time-consuming for users. To address this issue, concise representations of high-utility itemsets have been proposed such as closed HUIs, maximal HUIs and generators of HUIs. In this paper, we explore a novel representation called the minimal high utility itemsets (MinHUIs), defined as the smallest sets of items that generate a high profit, study its properties, and design an efficient algorithm named MinFHM to discover it. An extensive experimental study with real-life datasets shows that mining MinHUIs can be much faster than mining other concise representations or all HUIs, and that it can greatly reduce the size of the result set presented to the user.