## 2 \(L^{p}\) Averaging  
We present in Figures 1-3 the functionals \(A(a)\) for Distributions 1-3, respectively, for various \(p\). These functionals \(A(a)\) have one global minimum at or near \(r=0\), no additional minima, are convex in a neighborhood of the global minimum and are concave outside of this neighborhood. The fact that the \(A(a)\) are not globally convex is not important. Each \(A(a)\) is radially monotonically increasing outward from its minimum, which is sufficient to guarantee that there is only one global minimum and that there are no other local minima. On every finite closed interval in Figures 1-3 that does not include the global minimum, the derivative \(\mathrm{d}A/\mathrm{d}a\) is bounded away from 0. Hence, in all these cases, standard line-search methods converge to the global minimum in polynomial time. The structure of \(A(a)\) seen in Figures 1-3 is due to the fact that \(A(a)\) is based on a probability density function with strictly monotonically decreasing density in the radial directions outward from the mode. This structure does not generically occur for density functions \(f(r)\) and \(g(r)\) representing, for example, irregular scattered clusters. However, averaging in general and \(L^{p}\) averaging in particular make little sense when the data are clustered irregularly. The computational results presented in Figures 1-3 suggest the hypothesis  
The structure of the \(L^{p}\) averaging functional \(A(a)\) seen in Figures 1-3 and described in the previous paragraph occurs for all symmetric distributions, a situation that can be shown as follows. For symmetric distributions (that is, those for which \(g(r)=f(r)\)), the \(L^{p}\) averaging functional \(A(a)\) can be written as  
\[A(a)\;=\;\int_{0}^{\infty}(\;|r+a|^{p}+\;|r-a|^{p}\;)\;f(r)\;\mathrm{d}r \tag{9}\]  
\(A(a)\) is symmetric around \(a=0\), so we need consider only the behavior of \(A(a)\) for \(a\geq 0\). For \(a\geq 0\),  
\[\tfrac{\mathrm{d}A}{\mathrm{d}a}(a)\;=\;\int_{0}^{\infty}(\;(r+a)^{p}-\;|r-a|^ {p}\;)\;\Big{(}-\;\tfrac{df}{dr}(r)\;\Big{)}\;\mathrm{d}r \tag{10}\]  
and  
\[\frac{d^{2}A}{\mathrm{d}a^{2}}(a) \;=\;\int_{0}^{a}p\;(\;(r+a)^{p-1}-\;(\;a-r\;)^{p-1})\Big{(}- \frac{\mathrm{d}f}{\mathrm{d}r}(r)\;\Big{)}\;\mathrm{d}r \tag{11}\] \[+\int_{a}^{\infty}p\;(\;(r+a)^{p-1}+\;(\;r-a\;)^{p-1})\,\Big{(}- \tfrac{\mathrm{d}f}{\mathrm{d}r}(r)\;\Big{)}\;\mathrm{d}r\]  
One computes expressions (10) and (11) by differentiating the right sides of expressions (9) and (10), respectively, with respect to \(a\). One expresses the integral to be differentiated as the sum of an integral on (0,\(a\)) and an integral on (\(a\),\(\infty\)) and differentiates these two integrals separately. To simplify \(\mathrm{d}A\)/\(\mathrm{d}a\) to the form given in (10), one integrates by parts and combines the two resulting integrals. From these expressions, one obtains first that \(\mathrm{d}A\)/\(\mathrm{d}a(0)=0\) and \(\mathrm{d}^{2}A\)/\(\mathrm{d}a^{2}(0)>0\), that is, there is a local minimum at \(a=0\) and second that, for all \(a>0\), \(\mathrm{d}A\)/\(\mathrm{d}a(a)>0\), that is, \(A\) is strictly monotonically increasing for \(a>0\). Thus, for symmetric pdfs, \(A(a)\) has its global minimum at \(a=0\), that is, the \(L^{p}\) average exists and is equal to the mode of the distribution. There are no places where \(\mathrm{d}A\)/\(\mathrm{d}a=0\) other than at \(a=0\) and, on every finite closed interval that does not include the mode 0, \(\mathrm{d}A\)/\(\mathrm{d}a\) is bounded away from 0. Standard line-search methods for calculating the minimum of this \(A(a)\) are thus globally convergent. A general analytical structure for asymmetric distributions analogous to that described above for symmetric distributions is not yet available because, for asymmetric distributions, the properties of \(A(a)\) depend on additional properties of the probability density functions \(f(r)\) and \(g(r)\) that have not yet been clarified.