Mixed surface area measures

To each \((n-1)\) -tuple \((C_1,\ldots ,C_{n-1})\) of compact convex sets we may associate a Borel measure on \(\mathbb {S}^{n-1}\) which “represents” (by integration) the functional \(\mathcal{C}\mathcal{C}^n \ni L \mapsto V(L,C_1,\ldots ,C_{n-1})\) . This is the so-called mixed surface area measure of \(C_1,\ldots ,C_{n-1}\) , and the formal definition is given right below. The existence of such a measure will be derived from the Riesz representation theorem for measures.