0$. All that remains to be proved is the necessity of condition (iii). If $\log \overline{D} (h \of f)\not\in\op{BV}_{loc}((m,b))$, then no such function $q\in L^1$ can be found: there exists no $g_0$ for which $\log g_0'$ grows slower than $\log (h_1 \of f)'$ since $\log g_0'(z)$ would necessarily become unbounded before $z$ reached $h_1(f(b))$. \end{proof} \begin{remark} Theorem \ref{thm:1-d} shows that quasiconcavity is not quite equivalent to concavifiability. In addition, we require condition (ii), which roughly says that after straightening out one side, the other side has no horizontal or vertical tangencies. And beyond that, one still needs the yet more subtle condition (iii) governing the oscillation of the derivative on the unstraightened side. Strictly speaking, condition (ii) is superfluous in that it only serves to establish the existence of the function in condition (iii), where its existence is implicit. In particular, we need $\log \abs{\overline{D}(h\of f)}$ to exist almost everywhere in order to make sense of it being in $\op{BV}_{loc}$. Once it belongs to $\op{BV}_{loc}$ we can conclude that $h\of f$ on $(m,b)$, and its inverse, are locally Lipschitz. \end{remark} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \section{Functions on an Arbitrary Geodesic Metric Space} \label{sec:metric} We now extend our results to $\R^n$ and more general geodesic metric spaces. \begin{definition}\label{def:qc_metric} %(QUASICONCAVITY AND CONCAVITY ON A GEODESIC METRIC SPACE) A function $f \suchthat X\toarrow \R$ on a geodesic metric space is {\em (strictly/weakly) quasiconcave} if and only if $f \of \gamma \suchthat [0,1] \toarrow \R$ is (strictly/weakly) quasiconcave for every geodesic $\gamma \suchthat [0,1]\toarrow \R$, where we assume that $\ga$ is parameterized so that $d(\ga(s),\ga(t))=\abs{t-s}d(\gamma(0),\gamma(1))$ for all $s,t\in [0,1]$. %\addtocounter{footnote}{1}\footnotemark[\value{footnote}] \hspace{18pt}Similarly, $f$ is {\em (strictly/weakly) concave} if and only if for each geodesic $\ga \suchthat[0,1]\to X$, $f\of \ga$ is (strictly/weakly) concave as a function on $[0,1]$. (Note that this definition generalizes standard concavity for $X=\R^n$ with the Euclidean metric.) \end{definition} %\footnotetext[\value{footnote}]{ That is, we % assume that our geodesics are parameterized so that if $\ga(0)=x$ and $\ga(1)=y$ then %$d(\ga(s),\ga(t))=\abs{t-s}d(x,y)$ for all $s,t\in [0,1]$.} In what follows, we let $m\in X$ be the unique point maximizing $f$ if $f$ is quasiconcave or minimizing $f$ if $-f$ is quasiconcave. We will denote the negative part of $F$ by $F^{-}$ as in \eqref{eq:q-}. %Also recall that for a function %$F:X\to \R$, the negative part of $F$, $F^{-}$, is defined by %\[ %F^-(x) = \begin{cases} F(x) & F(x)<0\\ 0 & F(x)\geq 0\end{cases}. %\] We now state the complete criterion for concavification of quasiconcave functions, generalizing Theorem \ref{thm:1-d}. \begin{comment} %In what follows, let $\op{BV}_C$ denote the subset of $f\in\op{BV}([0,1])$ such %that $\op{Var}(f)\leq C$. %Following Remark \ref{rem:1-d} after Theorem \ref{thm:1-d}, for any constant $C>0$, we denote by $\Omega$ the family of strictly increasing functions $f \suchthat [0,1]\toarrow \R$ with \comment{Does $\overline{\abs{D}}(f)$ exist for arbitrary metric spaces, with no norm?} %\begin{enumerate} %\item[(i)] % $\log \overline{D}(f)$ bounded above on $[0,t]$ for any $t<1$ and bounded below on $[s,t]$ for any $0~~1$ the term $\inner{\grad f,e_j}$ identically vanishes, and so \[ f_{ij}=-\inner{\grad f,\nabla_{e_i}e_j}=-\norm{\grad f}\inner{e_1,\nabla_{e_i}e_j}=-\norm{\grad f}(\nabla_{e_i}\inner{e_1,e_j}-\inner{\nabla_{e_i}e_1,e_j})=\norm{\grad f}(\inner{\nabla_{e_i}e_1,e_j}). \] In particular $f_{ii}=-\norm{\grad f}\la_i$ when $i>1$. Putting this together we compute the Hessian of $g\of f$ to be, \begin{equation} \label{e21} \grad^2 (g \of f)=(g'' \of f)\begin{pmatrix} \norm{\grad f}^2& 0 & 0 & \cdots &0\\ 0& 0 & 0 & \cdots &0\\ 0& 0 & 0 & \cdots &0\\ \vdots & \vdots & \ddots & \vdots &\vdots\\ 0& 0 & 0 & \cdots &0 \end{pmatrix}+ (g' \of f)\begin{pmatrix} f_{11}& f_{12} & f_{13} &\cdots & f_{1n} \\ f_{21} & -\lambda_2 \norm{\grad f} & 0 & \cdots & 0 \\ f_{31} & 0 & -\lambda_3 \norm{\grad f}& \cdots &0 \\ \vdots & \vdots & \ddots & \vdots & \vdots \\ f_{n1} &0 &\cdots & 0 &-\lambda_{n}\norm{\grad f} \end{pmatrix}, \end{equation} Note we have $f_{1j}=-\inner{\grad f,\nabla_{e_1}e_j}$ for $j>1$, and moreover, $$ f_{11}=\frac{1}{\norm{\nabla f}^2} \inner{\nabla_{\nabla f}\nabla f,\nabla f}= \frac{\nabla_{\nabla f}\norm{\nabla f}^2}{2\norm{\nabla f} ^2}=\frac{\nabla_{\nabla f}\norm{\nabla f}} {\norm{\nabla f}}=\nabla_{e_1}\norm{\nabla f}, $$ or, in other words, $f_{11}$ is the growth rate of $\norm{\nabla f}$ in the $\nabla f$ direction. Similarly, since $\inner{e_1,e_1}=1$ identically, $\inner{e_1,\nabla_{e_j}e_1}=\frac12 e_j(\inner{e_1,e_1})= 0$. Therefore, \begin{equation} \label{e22} f_{1,j}=f_{j,1}=\nabla_{e_j}\inner{\grad f,e_1}-\inner{\grad f,\nabla_{e_j}e_1}=\nabla_{e_j} \norm{\nabla f}. \end{equation} Since the values $\la_i$ are all positive, we see that the principal minors, starting from the lower right, alternate sign. Hence in order to show that the eigenvalues of $\op{Hess}(g\of f)$ are all negative it remains to show that the sign of the entire determinant is $(-1)^n$. Observe that for $j>1$, the $(1,j)$-minor matrix of the combined matrix, formed by removing the first row and $j$-th column, can be made lower triangular by the following operations. We move the $j-1$-th row of the minor matrix, whose entry begins with $f_{j1}$, to the first row and shift all of the $j-2$ rows above the $j-1$-th row down by one. These operations introduce a $(-1)^{j-2}$ factor to the value of the minor, the determinant of the minor matrix, which is then $(-1)^{j-2}(g' \of f) ^{n-1} f_{j1}\lambda_2\dots\lambda_{j-1} \lambda_{j+1}\dots\lambda_n (-\norm{\grad f})^{n-2}$. In particular the $(1,j)$- entry times the $(1,j)$-cofactor for $j>1$, namely $(-1)^{j-1}f_{1j}(g'\of f)$ times the $(1,j)$-minor, is simply $$(-1)^{n-1}(g' \of f)^{n} \norm{\grad f}^{n-2} \frac{f_{1j}^2}{\lambda_j}\prod_{i=2}^n\lambda_i.$$ Hence the sum of these expressions from $j=1$ to $n$, corresponding to the determinant expansion across the first row, yields the entire determinant of $\op{Hess}(g \of f) $, namely \begin{equation} \label{e23} \det \op{Hess}(g \of f)=(-1)^{n-1}\left( \frac{g'' \of f}{g' \of f} \norm{\grad f}^2+f_{11}+\sum_{j=2}^n \frac{f_{1j}^2}{\lambda_j\norm{\grad f}}\right)(\norm{\grad f})^{n-1}(g' \of f)^{n}\prod_{i=2}^n \lambda_i. \end{equation} For $j~~0$ and $g''<0$, provided that for almost every value $t$ in the range of $f$, the quantity $$ \frac{1}{\norm{\nabla f}^2} \left(-f_{\s 11}-\sum_{j=2}^n \frac{f_{{\s 1}j}^2}{\lambda_j \norm{\grad f}}\right) $$ is bounded below by a value $q(t)>-\infty$ on the level set $f^{\scriptscriptstyle -1}(t) $, and where $q^{-}\in L^1_{loc}$. By the theorem's assumption we have such a bound. The $g$ function in the statement of the theorem then satisfies condition \eqref{e24}. Conversely, $\grad f$ must be bounded, away from the maximum point $m$, by the condition in Theorem \ref{thm:metric}. If we cannot find such a function $q$ then we cannot obtain a $g$ which everywhere satisfies the needed inequality, for the same reason as for the analogous result in the proof of Theorem \ref{thm:1-d}. \begin{comment} Since we assumed that the $C^1$ norm of $\nabla f$ is bounded, and the eigenvalues of the second fundamental form are uniformly bounded away from 0 (uniform quasiconcavity), it follows that each term $\frac{f_{1j}^2}{\lambda_j}$ is uniformly bounded. Hence if the gradient is uniformly positive on each level set, then we can solve for a smooth $g$. \end{comment} \end{proof} Theorem \ref{thm:manifold} generalizes the ``one-point'' conditions of Fenchel \cite{Fenchel} for $\R^n$ (as reformulated in Section 4 of \cite{Kannai:77}) to the Riemannian setting. Kannai's condition (I) on utility $v$ corresponds to our condition (ii) on $f$. However he is allowing for weak concavifiability, which accounts for his necessary conditions (II) and (III) differing from our condition (i) when the sublevel sets of $v$ are not strictly convex. Otherwise, these conditions are equivalent to our condition (i) and his conditions (IV) and (V) are folded into our condition (iii). This is best seen through the rephrasing of Kannai's condition (IV) as (IV$'$) and noting that his quantity ``$k$'' equals our $\norm{\grad f}$ and that under our assumptions in his setting when $M=\R^n$, we have $-\la_j \norm{\grad f}=f_{jj}$. Note also that Kannai's perspective is that of constructing a concave utility function based on weakly convex preference relations, whereas we start with an arbitrary function and see if it can be concavified. \begin{example}\label{ex:condition3} {\bf What condition (iii) excludes. } Condition (iii) can be easily violated by a $C^2$ function $f$ satisfying conditions (i) and (ii) by allowing for noncompact level sets which become asymptotically flat sufficiently quickly as points tend to infinity. A simple example is the quasiconcave function $f(x,y)=e^{e^x} y$ defined in the open positive quadrant of $\R^2$, shown in Figure \ref{fig:condition3}. (While this is not a $C^2$ manifold with boundary, we could smooth the corner to make it so.) Its gradient, $\grad f=\left(e^{x+e^x} y,e^{e^x}\right)$, is nonvanishing and its Hessian restricted to the level set of value $t$ as a function of the $x$ coordinate is $f_{22}=-\la_2 \norm{\grad f}=-\frac{t e^x \left(e^x-1\right)}{t^2 e^{2 x-2 e^x}+1}$. Similarly, $f_{11}=\frac{t e^{2 x} \left(t^2 e^{x-2 e^x} \left(e^x+1\right)+2\right)}{t^2 e^{2 x-2 e^x}+1}$ and $f_{12}=-\frac{e^{x+e^x} \left(t^2 e^x+e^{2 e^x}\right)}{t^2 e^{2 x}+e^{2 e^x}}$. The negative definiteness shows that $f$ is strictly quasiconcave. The quantity in condition (iii) on the level set of $t$ works out to be $\frac{1}{t(e^{-x}-1)}$, whose infimum over $x>0$ is always $-\infty$ for each $t$, and thus $f$ is not concavifiable. Intuitively, a concavifying $g$ must raise the values of the level sets along the $y$-axis, whereas much further along the $x$-axis on the same level sets $g$ must squash the function $f$. \end{example} \begin{figure}[h!] \centering \includegraphics[width=1.5in]{condition3} \caption{\sc A function violating condition (iii) of Theorem \ref{thm:manifold} and some level sets} \label{fig:condition3} \end{figure} \bigskip \begin{remark} Since $f_{1j}=-\inner{\grad f,\nabla_{e_1}e_j}$, in the special case that the integral curves of the vector field $\grad f$ lie along geodesics of $M$, then $f_{1j}=0$ for all $j>1$. This occurs, for instance, when $f$ is constant on distance spheres about a fixed point. In this case condition (iii) in Theorem \ref{thm:manifold} becomes vacuous. \end{remark} \bigskip \begin{remark} Some twice-differentiable functions $f$ with $\grad f$ vanishing at points other than the maximum can also be concavified, provided we are willing to concavify using a $g$ which is not twice differentiable. The more general condition is that after an initial postcomposition by a non-twice differentiable function $g_o$ the resulting $g_o\of f$ must satisfy conditions (ii) and (iii). In particular, when $\grad f$ vanishes at a point, it must do so on the entire level set, though this alone is not sufficient. \end{remark} \bigskip \begin{remark} % Remark 14 In contrast to Theorems \ref{thm:1-d} and \ref{thm:metric}, here $f$ is Lipschitz from the beginning, by virtue of being twice differentiable, and moreover $\log (h\of f\of \ga)'$ automatically belongs to $\op{BV}_{loc}$ for any twice differentiable increasing function $h$ and geodesic $\ga$ under the assumption of condition (ii). Also, condition (iii) of Theorem \ref{thm:manifold} is vacuous for one-dimensional $M$ and $C^2$ function $f$ when condition (ii) holds. So applying the theorem to one-dimensional examples is pointless. \end{remark} If $f$ is $C^2$ with nonvanishing gradient, then the quantity \eqref{e20} in the infimum of the definition of $q(t)$ in condition (iii) of Theorem \ref{thm:manifold} is uniformly bounded and continuous on compact sets. (Recall that the $\la_j$ are uniformly positive on the convex compact level sets.) Moreover, the infimum of any compact family of continuous functions is always continuous. Hence, we immediately obtain that the variation function $q$ from \eqref{e20} is continuous if $f$ is $C^2$ with compact level sets. We express this as the following especially simple corollary. \bigskip \begin{corollary}\label{cor:1} If $f \suchthat M\toarrow \R$ is strictly quasiconcave and $C^2,$ with compact level sets, then there is a $C^2$ strictly concavifying $g$ if and only if $\grad f$ does not vanish except possibly at $f$'s global maximum and minimum points, if any. \end{corollary} \begin{remark} Fenchel's Example from Figure \ref{quasifig-aumann.jpg} does not satisfy the conditions of Corollary \ref{cor:1}, because it is not strictly quasiconcave. In fact, for any function not strictly quasiconcave, at least one of the principal curvatures $\la_i$ vanishes somewhere and thus quantity \eqref{e20} becomes unbounded. \end{remark} %We can also work with the weak Hessian of $f$ for functions $f\in W^{2,1}$. Recall that a %weak gradient for $f \suchthat \Omega\subset\R^n\toarrow \R$ is any function $\phi %\suchthat \Omega\toarrow \R^n$ such that for every smooth compactly supported function %$\rho \suchthat \Omega\toarrow \R$, %\begin{equation}\label{e26} %\int_{\Omega}\phi(x) \rho(x)dx=-\int_\Omega f(x) (\grad %\rho)(x) dx. %\end{equation} % %We will denote any such weak gradient by $\nabla f$. This is justified, since from the %definition any two weak gradients agree almost everywhere. By taking charts and using the %volume forms one can see that this definition extends to arbitrary tensors on arbitrary %smooth manifolds. Define $W^{0,p}(M)$ to be $L^p(M)$, which we extend to denote the space %of $L^p$ tensors of any type on $M$. We then extend inductively by defining $f\in W^{k,p}(M)$ %if $\nabla f\in W^{k-1,p}(M)$. We only used $L^1$ existence of second derivatives in Theorem %\ref{thm:manifold}'s proof, though all inequalities must be understood in the weak sense of holding only after integrating against positive test functions. Thus, we obtain another corollary. %%Nah q(t) doesn't necessarily exist anywhere! % %\begin{corollary}\label{cor:2} %The result of Theorem \ref{thm:manifold} holds for $f\in W^{2,1}(M)$, provided we interpret condition (ii) and the conclusion in the weak sense. %\end{corollary} % %For $f\in W^{2,1}(M)$, taking small local convolutions yields smooth functions $f_\eps %\suchthat M\toarrow \R$ whose Hessians converge in $L^1$ to the weak Hessian of $f$. %Hence, the $W^{2,1}(M)$ Sobolev space provides a large and flexible class of functions %with good closure and embedding properties for which the ``differential'' conditions in %Theorem~\ref{thm:manifold} are still easy to check. % %%The space $W^{2,1}$ is a fairly large, and flexible, class frequently used in %%the theory of partial differential equations because of its closure properties and %%techniques for embedding it in other function spaces. %--------------------------------------------------------- % %\section{Concluding Remarks} % % We have tried in this article to clarify the relationship between %concavity and quasiconcavity at two levels. The first level is the %intuitive one, where we have explored when continuous strictly %quasiconcave functions, whether differentiable or not, are functions %that can be monotonically blown up into strictly concave functions. The %second level is the rigorous one, where we have shown that the intuition %holds true for arbitrary geodesic metric spaces of any dimension, but %subject to caveats that apply even in $\R$, caveats involving the %Lipschitz continuity of the function and the amount of variation in its %derivatives. %--------------------------------------------------------------- \bigskip %\section*{References} \begin{thebibliography}{AAA} \bibliographystyle{myalpha} %\bibliography{allbib} \bibitem{Aubin} Aubin, Thierry, {\it Nonlinear Analysis on Manifolds. Monge-Ampre equations}. 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