Complete sets in normed linear spaces

The concept of complete sets in normed linear spaces is a natural generalization of the concept of sets of constant width in Euclidean spaces. A bounded set in a normed linear space is said to be diametrically complete, diametrically maximal, or simply complete if it cannot be enlarged without increasing the diameter. We present a survey of results on complete sets in the following directions: elementary properties and characterizations; radii, centers, and diameter; relations to sets of constant width and balls; perfect spaces; completions of bounded sets; and uniqueness of completion.