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Towards a well-defined median. (English) Zbl 1140.41005
One considers the space \(\mathbb R^{n}\) \((n\geq2)\) endoved with \(l_p\)-norm, \(p\in[1,\infty]\). Then the diagonal \(\Delta_n= \{x= (x_1,x_2,\dots,x_n)\in\mathbb R^n: x_1=x_2=\dots= x_n\}\) is a proximinal subset of \(\mathbb R^n,\) and for \(p\in(1,+\infty]\) is even Chebyshev with respect to the \(l_p\)-norm. This means that the function \(f_p:\mathbb R^n\rightarrow\mathbb R,\) \(f_p(x)=\|(x,x,\dots,x)- (a_1,a_2,\dots, a_n)\|_p^s= \sum_{j=1}^n | x-a_j| ^{p}\) attains its minimum at a unique point \((\mu_p,\mu_p,\dots, \mu_p)\in \Delta_n\) if \(p\in(1,\infty]\), and the minimum point may be nonunique for \(f_1\) (i.e., for \(p=1\)). In fact for \(a_1\leq a_2\leq\dots \leq a_n\), \(f_1\) attains its minimum at the point \((a_{(n+1)/2},\dots, a_{(n+1)/2})\in \Delta_n\) if \(n\) is odd and at any point in the interval \([a_{n/2},a_{1+n/2}]\) if \(n\) is even. The authors proves that \(\lim_{p\rightarrow q}\mu_p= \mu_q,\) for all \(q\in(1,\infty].\) If \(p\in(1,\infty]\) and \(\mu_p= \mu_p(a),\) \(a=(a_1,a_2,\dots,a_n)\) is the minimum point for \(f_p,\) then there exists \(\lim\mu_p\) as \(p\) decrease to 1, and this limit, denoted by \(\mu^*\), is called by the authors the median of \(a=(a_1,\dots, a_n),\) or of the data set \(\{a_1,a_2,\dots,a_n\}\).

MSC:
41A50 Best approximation, Chebyshev systems
26E60 Means
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