A convex envelope is a mathematical concept that refers to the smallest convex shape that can enclose a set of points in a given space. This shape is important in optimization and computational geometry, as it helps in simplifying complex problems by focusing on the outer boundary of a dataset. In practical applications, convex envelopes are used in fields like computer graphics, data analysis, and machine learning. They assist in tasks such as shape recognition and clustering by providing a clear outline of the data's distribution, making it easier to analyze and interpret.