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One of the conditions of the central place theory is the assumption of a constant k parameter – a share of a central place in the population of the area served by this central place – for all levels of the Christaller’s hierarchy. Nevertheless, we did not find a rigorous proof of this assertion (underlying the Beckmann-Parr equation) in the bibliography on the central place theory. If this condition is assumed true, it also remains unclear – whether for all or only for strictly defined k-values. We have established that if the chosen K-value of the Christaller’s hierarchy is constant at every lattice level, the Beckmann-Parr equation holds for all meaningful values of k. At the same time we found that the range of k-values for an ideal Christaller’s lattice is bounded above by not an asymptote at k = 1, but an exact almost twice smaller value equal to K-(K^2-K)^0.5.  Since the latter changes very slightly during a radical rearrangement of the lattice from K = 3 to K = 7, we can state that we have discovered the new nonstrict invariant in the central place theory – the maximum value of k.

Is the share of a central place in the population of the area, served by this central place, a constant for all levels of the Christaller’s hierarchy?