Metric axiomatics of Euclidean space (Q1781283)

The author characterizes the Euclidean 3-space as a metric space whose metric satisfies a set of 11 axioms. He proves that the Euclidean metric on the Euclidean 3-space satisfies these axioms and that any metric space whose metric satisfies these axioms must necessarily satisfy Hilbert's axioms of the Euclidean 3-space. The axioms are quite natural and can be extended to define the Euclidean \(n\)-space for any \(n\). The question of independence is not addressed, and the author thinks it is conceivable that his system of axioms can be simplified. The author's axioms are axioms about the metric and as such they differ quite essentially from the metric axioms for plane geometry introduced by \textit{S. MacLane} [Am. Math. Mon. 66, 543--555 (1959; Zbl 0087.35102)] and from the metric axioms for space geometry introduced by \textit{R. Brossard} [Am. Math. Mon. 74, 777--788 (1967; Zbl 0168.18504)].