A Theorem on Convex Sets with Applications

Abstract

T. W. Anderson [1] has proved the following theorem and has given applications to probability and statistics. THEOREM 1. Let $E$ be a convex set in $n$-space, symmetric about the origin. Let $f(x) \geqq 0$ be a function such that i) $f(x) = f(-x)$, ii) $\{x |f(x) \geqq u\} = K_u$ is convex for every $u (0 \geqq u \geqq \infty)$ and iii) $\int_E f(x) dx < \infty$, then$ \begin{equation*}\tag{(1)}\int_E f(x + ky) dx \geqq \int_E f(x + y) dx\quad for 0 \leqq k \leqq 1.$ The purpose of this paper is to prove what can be considered a generalization of Anderson's Theorem and to give different statistical applications. Functions in $L_1$ satisfying the hypothesis were called unimodal by Anderson and he noted in [1] that if we let $\varphi(y)$ be equal to the right hand side of (1) then $\varphi$ is not unimodal in his sense insofar as it does not necessarily satisfy $ii$ (i.e., there exist $f, E$, and $u$ such that $\{x\mid \varphi(x) \geqq u\}$ is not convex). His example is the case where $n = 2$ and \begin{equation*}f(x) = \begin{cases}3,\quad\|x_1\| \leqq 1,\quad\|x_2\| \leqq 1, \\ 2,\quad\|x_1\| \leqq 1,\quad 1 < \|x_2\| \leqq 5, \\ 0,\quad {other} x,\end{cases}\end{equation*} where $x_1, x_2$ are the components of $x$ relative to rectangular cartesian coordinate system. Let $E$ be the set of vectors where $|x_1| \leqq 1, |x_2| \leqq 1$. The set $\{x \mid \varphi(x) \geqq 6\}$ is not convex since for $x = (.5,4)$ and $x = (1,0), \varphi(x) = 6$, while for $x = (.75.2), \varphi (x) < 6$. The point of departure of this paper is to see what can be said about $\varphi$. This is achieved in Theorem 2, giving a stronger and more symmetrical statement than Anderson makes (but one which does not yield more information for his applications). The main Lemma, presented below, proceeds along the line of his argument but squeezes out additional information (convexity of level lines) under a weaker hypothesis (no symmetry assumptions) than Anderson uses at the corresponding stage of his argument.