Separating hyperplanes and support functions

Let C be a nonempty closed convex subset of $\mathbb R^d$, and let x lie outside C. A nearest point p exists: a minimizing sequence can be restricted to a bounded closed region, where compactness yields a convergent subsequence. Convexity gives uniqueness through the strict convexity of squared distance. For any z in C, move from p a small distance toward z. The squared distance to x cannot decrease, so its one-sided derivative is nonnegative. The resulting inner-product inequality identifies the normal direction a=x−p.

Every feasible z then has score at most the score of p, whereas x has an additional squared-distance margin. This strictly separates the outside point from the closed convex set, even if the set is unbounded. It is stronger than the general weak separation statement for two disjoint nonempty convex sets. Uniform positive-margin separation between arbitrary disjoint convex sets needs additional hypotheses. For example, the intervals (−1,0) and (0,1) are disjoint and admit a pointwise-strict separating level at zero, but there is no positive gap between their closures.

For a nonempty set C, its support function at a is the supremum of $\langle a,z\rangle$ over z in C. It may be positive infinity when the set extends indefinitely in that direction. Taking the supremum makes the support function convex and positively homogeneous for nonnegative scaling. It does not require C itself to be convex. Indeed, replacing C by its closed convex hull leaves every support value unchanged. Linear measurements cannot distinguish a set from all of its convex mixtures and limit points.

For the Euclidean ball with center c and radius r, the support is $\langle a,c\rangle+r\|a\|$. Cauchy–Schwarz gives the upper bound, and moving from c in direction a attains it when a is nonzero. For the nonnegative ray, a positive direction gives infinity while a nonpositive direction gives zero. Infinite values are meaningful geometric information, not a failed calculation. Translation shifts the support by a linear term, and scaling the set by a nonnegative factor scales its support wherever the extended-value operation is defined.

A nonempty closed convex set equals the intersection of all halfspaces specified by its support function. One inclusion is immediate: every member satisfies every upper bound. For the converse, an outside point has the strict separator just constructed, so it violates at least one bound. Closedness matters because a halfspace intersection is closed. Convexity matters because a halfspace intersection is convex. Applied to an arbitrary nonempty set, the same construction returns its closed convex hull, potentially filling gaps and adding missing boundary points.

Supporting hyperplanes also describe boundary contact. For a closed convex set with nonempty interior, each boundary point admits a nonzero supporting normal; one can prove this by approaching from outside and taking a convergent sequence of normalized separating directions. The limit gives contact but need not give a positive margin. For a lower-dimensional set, one should distinguish support in the ambient space from support relative to its affine hull. A normal perpendicular to the whole affine hull is valid but says nothing about which relative boundary face is touched.