Inspheres and inner products
Let X be a normed linear space of dimension \(\geq 3\), with unit ball B. The main authors' aim is to provide the following characterization results: A) \(X\) is a finite dimensional Hilbert space iff for each convex body \(C\) in \(X\) and each maximum radius ball \(z+\rho B\) of \(C\), \(z\in Co(\partial (z+\rho B)\cap \partial C);\) B) If, in addition, \(x\) is reflexive, then, it is a Hilbert space iff for each convex body \(C\) in \(X\) and each maximum radius ball \(z+\rho B\) of \(C\), \(z\in cl\) \(co((z+\tau B)\cap \partial C)\) for all \(\tau >\rho.\) These, and other slightly sharper characterizations give an appropriate answer to a conjecture of \textit{C. Franchetti} and \textit{P. L. Papini} [''Approximation properties of sets with bounded complements'', Proc. R. Soc. Edinb. Sect. A 89, 75-86 (1981; Zbl 0479.41032)].
scientific article; zbMATH DE number 3975625
scientific article