## 1 Introduction
The classes of \(L^{p}\) and \(l^{p}\) minimization principles that we will investigate in this paper are those that represent univariate continuum \(L^{p}\) averaging and discrete \(l^{p}\) averaging, defined as follows. Univariate\(L^{p}\) and \(I^{p}\) averages are the real numbers \(a\) at which the following functionals \(A\) and \(B\) achieve their respective global minima:  
\[A(a):=\ \int_{-\infty}^{\infty}\lvert x-a\rvert^{p}\ \psi(x)\ \mathrm{d}x \tag{1}\]  
where \(\psi\) is a probability density function (pdf) that satisfies the conditions given below, and  
\[B(a):=\ \sum_{i}\lvert x_{i}-a\rvert^{p} \tag{2}\]  
where the \(x_{i}\) are data points from the distribution with pdf \(\psi\). The pdf \(\psi\) is assumed to have measurable second derivative and to satisfy the following two conditions:  
* radially strictly monotonically decreasing outwards from the mode (3a) \[\psi\ \mathrm{and}\ \mathrm{d}\psi/\mathrm{d}x\ \mathrm{bounded}\ \mathrm{by}\ c\lvert x\rvert^{\beta}\ \mathrm{and}\ c\lvert x\rvert^{\beta-1}\ \mathrm{, respectively, for given \(c\) and \(\beta>p+1\) as \(x\to\pm\infty\)  
Without loss of generality, we assume that the mode, that is, the \(x\) at which \(\psi\) achieves its maximum, is at the origin.  
In a departure from the traditional use of \(x\) as the independent variable of a univariate pdf, we will express univariate pdfs in radial form with \(r\) being the radius measured outward from the mode of the distribution. (This notation is chosen to allow natural generalization to higher dimensions in the future.) With the notation \(g(r)=\psi(\
eg r)\) and \(f(r)=\psi(r)\), \(r\geq 0\), functional \(A\) can be rewritten in the form  
\[A(a)\ =\ \int_{0}^{\infty}\lvert r+a\rvert^{p}\ g(r)\ \mathrm{d}r\ +\ \int_{0}^{\infty}\lvert r-a\rvert^{p}\ f(r)\ \mathrm{d}r \tag{4}\]  
Since functional (4) is finite only when  
\[p<\ \beta-1 \tag{5}\]  
the mean (\(L^{2}\) average) does not exist for distributions with \(\beta\leq 3\) and even the median (\(L^{1}\) average) does not exist for distributions with \(\beta\leq 2\). For example, the median does not exist for the Student \(t\) distribution with one degree of freedom because \(\beta=2\) for this distribution. To create meaningful \"averages\" in these cases, weighted and trimmed sample means have been proposed with success [21]. However, weighted and trimmed sample means require _a priori_ knowledge of the specific distribution and/or of various parameters, knowledge that is often not available. Minimization of the \(L^{p}\) functional (4) or of the \(I^{p}\) functional (2) is, when \(0<p<\min\{1,\ \beta-1\}\), an alternative for creating an \"average\" for a heavy-tailed distribution or of a sample thereof. In the present paper, we will investigate whether, by providing only the information that the data come from a \"standard\" statistical distribution that satisfies Conditions (3), the \(L^{p}\) and \(I^{p}\) averaging functionals \(A\) and \(B\) can be minimized in a way that leads to polynomial-time minimization of general \(L^{p}\) and \(I^{p}\) functionals. Specifically, in the next two sections, we will investigate to what extent the \(L^{p}\) and \(I^{p}\) averaging functionals are devoid of local minima other than the global minimum, a key feature in this process. For illustration of the theoretical results, we will present computational results for the following three types of distributions:  
_Distribution 1_: Gaussian (light-tailed distribution) distribution with probability density function  
\[f(r)=g(r)=\tfrac{1}{\sqrt{2\pi}}\exp\left(\tfrac{-r^{2}}{2}\right) \tag{6}\]  
_Distribution 2_: Symmetric heavy-tailed distribution with probability density function \[f(r)=g(r)=\tfrac{1}{c}\left[\left(1+\tfrac{\alpha}{2}\right)-\tfrac{ \alpha}{2}r^{2}\right]\,,0\ \leq r\leq 1\] (7a) \[f(r)=g(r)=\tfrac{1}{c}r^{-\alpha}\,,1<r<\infty\] (7b) where \[c=2\left[\tfrac{\alpha}{\alpha-1}+\tfrac{\alpha}{3}\right]\] (7c) (For Distribution 2, the \(\beta\) of condition (3b) is \(\alpha\).)