The Minkowski sausage is a geometric construct used in the field of topology and geometric measure theory, particularly in the study of the properties of sets in Euclidean space. Specifically, it refers to a way of "thickening" a curve in Euclidean space to create a three-dimensional shape. Given a continuous curve \( C \) in three-dimensional space, the Minkowski sausage is formed by taking a tubular neighborhood around the curve.
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The Minkowski sausage replaces a segment by eight directed unit segments whose endpoint displacement is four units. A middle straight run of length two represents two collinear unit segments. After normalizing the displacement to one, the eight contracting Euclidean similarities all have ratio . The open diamond is an open set condition witness: its eight image diamonds have disjoint interiors and lie within it. Hence , giving Hausdorff dimension . Treating the middle run as a single similarity of ratio instead defines a different iterated scheme.