The Fermat-Weber problem and inner product spaces
Let \(\{x_ i\}^ n_{i= 1}\) be points in a real Banach space \(X\). Let \(w_ i> 0\). For \(x\in X\), let \[ F(x)= \sum^ n_{i= 1} w_ i\| x- x_ i\|. \] Let \(m= \min\{F(x): x\in X\}\) and let \(M= \{x: F(x)= m\}\). \(M\) is always convex, and if \(X\) is a Hilbert space, then \(M\cap\text{conv}\{x_ i\}^ n_{i= 1}\neq\emptyset\). (Where \(\text{conv }K\) is the convex hull of \(K\).) This paper shows that if \(M\cap \text{conv}\{x_ i\} ^ n_{i= 1}\neq \emptyset\) for all finite sets \(\{x_ i\}^ n_{i= 1}\), then \(X\) is an innerproduct space.
scientific article; zbMATH DE number 617501