Minkowski sums and mixed volumes

For sets K and L, their Minkowski sum contains all x + y with x in K and y in L. It is different from the union, which contains the original points, and from the intersection, which contains shared points. Minkowski addition preserves convexity and compactness for nonempty compact convex sets. A positive scaling multiplies every point, and translation of one summand simply translates the entire sum, leaving its Euclidean volume unchanged.

Support functions linearize this addition. Maximizing a linear functional over x + y separates into independent maxima over x and y, so the support of K + L is the sum of their supports. This equality provides both a proof technique and a practical way to recognize resulting shapes. For example, adding two coordinate rectangles adds their corresponding side lengths, while adding two nonparallel line segments creates a parallelogram rather than another segment.

For convex bodies in real d-space, the volume of a nonnegative linear combination is a homogeneous polynomial of degree d in the scaling parameters. We use the standard convention that the mixed volume of d identical copies of K equals its ordinary d-dimensional volume. For two bodies the binomial coefficient accompanies each mixed coefficient. In the plane this gives area(K + tL) equal to area(K), plus twice t times the mixed area, plus t squared times area(L).

Mixed volume is symmetric in its d arguments and linear in each under Minkowski addition and nonnegative scaling. It is nonnegative, translation invariant in each argument, and continuous under suitable convex-body convergence. These are structural theorems, not properties obtained by integrating pointwise products of the two shapes. For example, two bodies can be disjoint after translation yet retain exactly the same mixed volume, proving it cannot measure ordinary overlap.

We use convex body to mean compact, convex, and with nonempty interior in the ambient space, so its volume is positive. The volume polynomial and mixed volumes extend to nonempty compact convex sets of smaller dimension by continuity, but some coefficients or leading terms may vanish. One must not infer positive volume for each summand merely because their sum has positive volume: two transverse planar segments already give a counterexample.

For polytopes, Minkowski addition combines their supporting directions and normal information, making the operation useful in motion planning and sparse-polynomial geometry. The volume polynomial does not by itself enumerate faces or construct the sum efficiently in high dimension. Distinguish a structural identity from an algorithm, and specify whether a subsequent inequality requires positive full-dimensional volume. The next lesson uses that positive-volume setting to relate mixed geometry to concavity of a volume root.