The study of convex sets, their convex hulls, supporting closed half-spaces, and inequalities for their sizes.
The Minkowski sum of subsets of a vector space is . Minkowski addition also makes sense for nonconvex sets; translations and positive dilations commute with it.
In , the sets and are closed because their points have no finite accumulation point. Their Minkowski sum contains but does not contain zero, since has no positive-integer solution. Compactness of one summand is a useful sufficient condition missing from this counterexample.
In Euclidean space, a Minkowski sum of a compact set and a closed set is closed. From any convergent sequence , extract a convergent subsequence of using compactness. The corresponding then converges to the difference of the two limits, which lies in . If is also bounded, the sum is bounded and therefore compact.
For nonempty open sets , the Lebesgue measure of their Minkowski sum satisfiesIt also holds for compact sets and in standard measurable-set formulations with the appropriate measurability qualification. Normalize both volumes to one and apply the Prékopa–Leindler inequality to indicator functions.
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Convex geometry is a branch of mathematics that studies convex sets and their properties in various dimensions. A set is defined as convex if, for any two points within the set, the line segment connecting those two points lies entirely within the set. This simplicity in definition leads to rich geometric and combinatorial properties.