A convex body is a specific type of geometric figure in Euclidean space that possesses certain characteristics. Formally, a convex body can be defined as follows: 1. **Compactness**: A convex body is a compact set, meaning it is closed and bounded.
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A convex body in is a compact convex set with nonempty interior in the ambient Euclidean space. It consequently has positive finite Lebesgue measure. A filled rectangle with positive side lengths and a closed positive-radius Euclidean disk are planar convex bodies. A line segment in has empty ambient interior and is not a convex body under this convention. This is the convention used in the Brunn-Minkowski theorem. [ref]
https://sites.stat.washington.edu/jaw/COURSES/520s/523/HO.523.20/Gardner.BullAMS.02.pdf. Gardner (2002), introduction