## 4 Conclusions  
The wide-spread impression that minimization of \(L^{p}\) and \(l^{p}\) functionals, \(0<p<1\), is combinatorially expensive is valid for general situations in which no structure of the data is known. However, the results in this paper suggest that, when the data come from an appropriate statistical distribution, \(L^{p}\) and \(l^{p}\) averages can be calculated in polynomial time. The approach of the paper is applicable without precise knowledge of the parameters of the distribution. One does not need precise knowledge of the parameters but rather only generalizations of Conditions (3), an upper bound on the exponent \(-\beta\) of the tail density and additional conditions for asymmetric distributions and for setting up a rule for choosing \(q\) in Algorithm 1.  
Topics for future research include  
* Quantitative rules for using information about the underlying continuum distribution to choose the \(q\) of Algorithm 1 based on a user's preferred tradeoff between maximum accuracy and minimum computational burden
* Investigation of the advantages and disadvantages of introducing smoothing in the \(B(x_{k})\) calculated in Step 4 of Algorithm 1 to increase the robustness against shallow local minima; connection of the smoothing with properties of the underlying distributions
* Description of the class(es) of symmetric and asymmetric univariate and multivariate distributions for which radially strictly monotonic \(L^{p}\) averaging functionals and radially nearly strictly monotonic \(l^{p}\) averaging functionals can be created and thus for which \(L^{p}\) and \(l^{p}\) averages can be calculated in polynomial time
* Investigation of convergence of the \(l^{p}\) average to the \(L^{p}\) average and of related issues of efficiency, optimality, breakdown point, influence function, _etc._
* Investigation of the conditions under which \(L^{p}\) and \(l^{p}\) averages converge to the mode as \(p\to 0\)
* Treatment of more general univariate and multivariate \(l^{p}\) minimization problems including but not limited to \(l^{p}\) regression and matrix-constrained \(l^{p}\) minimization, for example, minimization of \[\sum_{i=1}^{l}|x_{i}|^{p}\text{ subject to }A\mathbf{x}=\mathbf{b}\] (12) (cf. [17,18]) (The \(l^{p}\) averaging process considered in the present paper can be expressed in format (12).)  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \(\alpha\)\({}^{p}\) & 0.5 & 0.1 & 0.02 \\ \hline
3 & 0.021 & 0.015 & 0.015 \\
2 & 0.021 & 0.020 & 0.020 \\
1.5 & 0.031 & 0.029 & 0.029 \\
1.1 & 0.498 & 0.108 & 0.108 \\
1.05 & 2.56 \(\times 10^{29}\) & 0. 207 & 0.207 \\
1.04 & 1.14 \(\times 10^{45}\) & 0.257 & 0.257 \\
1.03 & 2.83 \(\times 10^{74}\) & 0.341 & 0.341 \\
1.02 & 1.52 \(\times 10^{119}\) & 2.37 \(\times 10^{6}\) & 0.516 \\ \hline \end{tabular}
\end{table}
Table 4: Sample \(l^{p}\) averages calculated by Algorithm 1 with window size \(2q+1=25\) for 2000-point data set from Distribution 2.  
Many phenomena in human-based areas (sociology, cognitive science, psychology, economics, human networks, social media, _etc._) are increasingly known to be represented by data that have large numbers of outliers and belong to very heavy-tailed distributions, which suggests that \(L^{p}\) and \(l^{p}\) averaging, \(L^{p}\) and \(l^{p}\) regression and more general \(L^{p}\) and \(l^{p}\) minimization tasks, \(0<p<1\), will be important in practice. The results of the present paper provide the first indication that one may be able to solve, in polynomial time, generically combinatorially expensive \(L^{p}\) and \(l^{p}\) minimization problems for these phenomena by requiring only \"natural\" statistical structure without having to impose restrictions such as sparsity and without having to accept suboptimal local solutions instead of optimal global solutions.