Abstract
We solve the problem of finding all $$(n+2)$$ -dimensional geometries defined by a nondegenerate analytic function $$ \varphi (\varepsilon _1x^1_Ax^1_B+ \cdots +\varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B,w_A,w_B),$$ which is an invariant of a motion group of dimension $$(n+1)(n+2)/2$$ . As a result, we have two solutions: the expected scalar product $$\varepsilon _1x^1_Ax^1_B+ \cdots +\varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B+\varepsilon w_Aw_B $$ and the unexpected scalar product $$\varepsilon _1x^1_Ax^1_B+ \cdots +\varepsilon _{n+1}x^{n+1}_Ax^{n+1}_B+w_A+w_B $$ . The solution of the problem is reduced to the analytic solution of a functional equation of a special kind.
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