Separation theorems translate convex geometry into linear inequalities. They explain why dual variables, supporting hyperplanes, and weak closedness are unavoidable in analysis and optimization.
Minkowski functionals
Minkowski functionals turn convex neighborhoods into sublinear functions. This converts geometric separation into Hahn-Banach domination.
Definition. Let \(C\) be a convex absorbing subset of a real vector space \(X\) with \(0\in C\). Its Minkowski functional is \(p_C(x)=\inf\{t>0:x\in tC\}\).
Lemma. If \(C\) is convex and absorbing, then \(p_C\) is sublinear. If \(C\) is open in a normed space and \(0\in C\), then \(C=\{x:p_C(x)<1\}\).
Proof. Positive homogeneity follows directly from the definition. For subadditivity, if \(x\in aC\) and \(y\in bC\), then \(x+y=(a+b)(a(a+b)^{-1}x/a+b(a+b)^{-1}y/b)\in(a+b)C\) by convexity; take infima over \(a,b\).
If \(p_C(x)<1\), then \(x\in tC\) for some \(t<1\), and since \(0\in C\), convexity gives \(tC\subset C\). Conversely, if \(x\in C\) and \(C\) is open, then \((1+\varepsilon)x\in C\) for sufficiently small \(\varepsilon>0\), so \(x\in(1+\varepsilon)^{-1}C\) and \(p_C(x)<1\).
Separation of a point from an open convex set
Separating a point from an open convex set is the basic hyperplane theorem. It says that convex nonmembership can be certified by one continuous linear functional.
Theorem. Let \(C\) be a nonempty open convex subset of a real normed space \(X\), and let \(x_0\notin C\). Then there exist \(f\in X^*\setminus\{0\}\) and \(\alpha\in\mathbb R\) such that \(f(x)<\alpha\le f(x_0)\) for every \(x\in C\).
Proof. Translate so \(0\in C\). Let \(p=p_C\). Since \(x_0\notin C\), \(p(x_0)\ge1\). On the line \(M=\mathbb R x_0\), define \(g(tx_0)=t\). For \(t>0\), \(g(tx_0)=t\le t p(x_0)=p(tx_0)\); for \(t<0\), \(g(tx_0)=t\le p(tx_0)\) because \(p\ge0\).
Hahn-Banach gives \(f\le p\) on \(X\) and \(f(x_0)=1\). If \(x\in C\), then \(p(x)<1\), so \(f(x)\le p(x)<1=f(x_0)\). Returning from the translation gives the stated inequality with the corresponding affine level \(\alpha\). Continuity of \(f\) follows because \(C\) contains a norm ball around \(0\), so \(p\) is bounded above by a multiple of \(\|\cdot\|\).
Closed convex sets are weakly closed
Closed convex sets are determined by their continuous linear inequalities. This is why weak and norm closures agree for convex sets.
Corollary. If \(C\subset X\) is convex and norm closed, then \(C\) is weakly closed.
Proof. Let \(x_0\notin C\). Since \(C\) is closed, there is an open convex neighborhood \(U=C+B(0,r)\) not containing \(x_0\) for some \(r>0\). By the separation theorem, some \(f\in X^*\) and \(\alpha\) satisfy \(f(u)<\alpha\le f(x_0)\) for all \(u\in U\). In particular \(f(c)<\alpha\le f(x_0)\) for \(c\in C\). Hence \(X\setminus C\) is a union of weakly open half-space neighborhoods, so \(C\) is weakly closed.
Supporting functionals
A supporting functional touches a convex set at a boundary point without cutting through it. This is the analytic form of a tangent hyperplane.
Proposition. Let \(C\subset X\) be open, convex, and nonempty. If \(x_0\in\partial C\), then there exists \(f\in X^*\setminus\{0\}\) such that \(f(x)\le f(x_0)\) for all \(x\in C\).
Proof. Since \(C\) is open and \(x_0\in\partial C\), one has \(x_0\notin C\). Apply the separation theorem for a point outside an open convex set. There are \(f\in X^*\setminus\{0\}\) and \(\alpha\) such that \(f(x)<\alpha\le f(x_0)\) for all \(x\in C\). Hence \(f(x)\le f(x_0)\) on \(C\).
Strict separation with positive distance
Positive distance allows strict separation because the separating functional can be normalized with a margin.
Proposition. If \(C\subset X\) is closed convex and \(\operatorname{dist}(x_0,C)>0\), then there are \(f\in X^*\), \(\|f\|=1\), and \(\alpha\in\mathbb R\) such that \(f(c)\le\alpha< f(x_0)\) for all \(c\in C\).
Proof. Let \(r=\operatorname{dist}(x_0,C)\) and set \(U=C+B(0,r/2)\). Then \(U\) is open, convex, and \(x_0\notin U\). Separate \(x_0\) from \(U\): \(f(u)<\beta\le f(x_0)\) for all \(u\in U\). If \(c\in C\) and \(\|h\|<r/2\), then \(c+h\in U\), so \(f(c)+f(h)<\beta\). Taking the supremum over \(\|h\|<r/2\) gives \(f(c)+(r/2)\|f\|\le\beta\). After normalizing \(f\), put \(\alpha=\beta-r/2\). Then \(f(c)\le\alpha<\beta\le f(x_0)\).
Convex dual certificates
Dual certificates are separation theorems written in optimization language. They prove impossibility or optimality by exhibiting a functional inequality.
Example. In optimization, separation says that infeasibility of \(x_0\in C\) can be certified by a continuous linear functional. This is the geometric origin of Lagrange multipliers, Fenchel duality, supporting hyperplanes, and many minimax theorems.