Generalized inverses of strictly monotone transformations

Abstract: Let $C \subseteq \mathbb{R}ⁿ$ be a convex set. The mapping $f : C \longrightarrow \mathbb{R}ⁿ$ is strictly increasing, if $ \langle f(x)-f(y), x-y \rangle > 0 $ for all distinct elements $x,y \in C$. Applying classical theorems of finite dimensional convex geometry and convex analysis, we show that the inverse functions has a unique extension $$f^{(-1)} : \mathrm{conv} (f(C)) \longrightarrow C$$ such that $f^{(-1)}$ is monotone, continuous and it acts as a left-inverse of $f$. As an application we introduce the concept of vector-valued weighted quasi-arithmetic means and discuss their equality problem.

• The paper establishes that every strictly monotone map on a closed convex subset of ℝⁿ has a unique, explicitly defined, continuous monotone left-inverse on the convex hull of its image.
• The results show that domain closedness is essential in higher dimensions, with an open half-disk and complex-square map providing a counterexample where no monotone extension exists.
• The paper defines vector-valued weighted quasi-arithmetic means and proves that on ℝⁿ they equal the arithmetic mean exactly when the generator is affine with a positive-definite linear part.

This paper by Péter Tóth answers an open problem posed by Zsolt Páles at the 60th International Symposium on Functional Equations concerning the higher-dimensional analogue of a classical one-dimensional fact: that the inverse of a strictly monotone (not necessarily continuous) function on an interval admits a unique monotone and continuous extension to the convex hull of its image — a construction whose most familiar instance is the quantile function. The paper establishes that for strictly monotone mappings on closed convex subsets of $\mathbb{R}^n$, such an extended monotone left-inverse exists, is unique, and is continuous, while demonstrating that closedness of the domain is essential. As an application, the author defines vector-valued weighted quasi-arithmetic means (QAMs) via the generalized inverse and characterizes when such a mean coincides with the weighted arithmetic mean.

Existence and uniqueness of the generalized inverse

The setting is a convex set $C \subseteq \mathbb{R}^n$ with a strictly increasing mapping $f : C \to \mathbb{R}^n$, meaning $\langle f(x)-f(y), x-y\rangle > 0$ for distinct $x,y$; such $f$ is automatically injective, so the ordinary inverse exists but its image domain need not be convex. The central question is whether $f^{-1}$ admits a monotone extension $f^{(-1)}$ to $\operatorname{conv}(f(C))$ acting as a left-inverse, and whether it can be continuous.

The construction proceeds through a necessary condition: any extended monotone left-inverse must satisfy

$$f^{(-1)}(z) \in \bigcap_{x \in C} \{ s \in C : \langle x-s, f(x)-z \rangle \geq 0 \},$$

i.e., $C \subseteq \mathbb{R}^n$0 must lie in every member of a family of closed half-space intersections with $C \subseteq \mathbb{R}^n$1. The paper then shows this intersection is nonempty — and in fact a singleton — via a chain of classical convex-geometric tools. A purely linear-algebraic lemma, proved using Ky Fan's minimax theorem, guarantees that every subfamily of $C \subseteq \mathbb{R}^n$2 of these sets intersects. Helly's theorem then upgrades this finite intersection property to a full intersection, with the boundedness hypothesis verified through a recession-cone argument: when $C \subseteq \mathbb{R}^n$3 is unbounded, a compactness/open-cover argument on the normalized recession directions produces a finite subfamily whose intersection is bounded. Uniqueness of $C \subseteq \mathbb{R}^n$4 follows from a short argument exploiting the strict monotonicity of $C \subseteq \mathbb{R}^n$5 on two interior points $C \subseteq \mathbb{R}^n$6 and $C \subseteq \mathbb{R}^n$7.

The main theorem states that for closed convex nonempty $C \subseteq \mathbb{R}^n$8 and strictly increasing $C \subseteq \mathbb{R}^n$9, there exists a unique increasing $f : C \to \mathbb{R}^n$0 with $f : C \to \mathbb{R}^n$1. A notable strength of the proof is that it is constructive: the generalized inverse is given explicitly by

$f : C \to \mathbb{R}^n$2

Moreover, if $f : C \to \mathbb{R}^n$3 is compact and convex, the construction extends $f : C \to \mathbb{R}^n$4 to the entire space $f : C \to \mathbb{R}^n$5 while preserving monotonicity, since Helly's boundedness condition holds for arbitrary $f : C \to \mathbb{R}^n$6 in the compact case. This is a strictly stronger conclusion than the one-dimensional result, which only guarantees extension to the convex hull of the image.

Closedness of the domain is essential

In dimension one, convexity of the domain alone suffices; the paper shows this fails in higher dimensions with an example attributed to K. Okamura. Let $f : C \to \mathbb{R}^n$7 be the open half-disk $f : C \to \mathbb{R}^n$8 and $f : C \to \mathbb{R}^n$9, which corresponds to the complex square map $\langle f(x)-f(y), x-y\rangle > 0$0. This $\langle f(x)-f(y), x-y\rangle > 0$1 is strictly increasing, but admits no extended monotone left-inverse. The image is the open unit disk minus the nonpositive real axis, so its convex hull is the full open unit disk; evaluating the necessary intersection condition at the origin yields the strict inequality $\langle f(x)-f(y), x-y\rangle > 0$2, a contradiction. The paper also notes the related fact, proved at the ISFE meeting, that no continuous extension of the ordinary inverse to $\langle f(x)-f(y), x-y\rangle > 0$3 exists. This contrast with the scalar case is a substantive structural finding: the closedness assumption is not a technical convenience but a genuine hypothesis.

Continuity of the generalized inverse

The second main theorem establishes that on a closed convex domain the generalized inverse is continuous. The proof is by contradiction: if $\langle f(x)-f(y), x-y\rangle > 0$4 fails to converge to $\langle f(x)-f(y), x-y\rangle > 0$5 along a sequence $\langle f(x)-f(y), x-y\rangle > 0$6, compactness of the unit sphere yields a direction $\langle f(x)-f(y), x-y\rangle > 0$7 along which the iterates escape at a fixed distance $\langle f(x)-f(y), x-y\rangle > 0$8. A geometric lemma shows that if a sequence of unit-normalized vectors has inner product tending to 1 with $\langle f(x)-f(y), x-y\rangle > 0$9, then shifting by a fraction of $x,y$0 along $x,y$1 preserves this limiting behavior. Choosing a point $x,y$2 on the segment between $x,y$3 and $x,y$4, strict monotonicity of $x,y$5 gives $x,y$6, and the defining half-space inequalities for $x,y$7 then force a violation of the monotonicity of $x,y$8 for large $x,y$9. The conclusion is that $f$0 is continuous on $f$1 — matching the one-dimensional behavior, but now under the closedness hypothesis that the Okamura example shows cannot be dropped.

Application: vector-valued quasi-arithmetic means

Using the generalized inverse, the paper defines the $f$2-variable weighted vector-valued quasi-arithmetic mean generated by a strictly monotone $f$3:

$f$4

This generalizes the classical scalar QAM and improves upon Leonetti's vector-valued variant, which assumes a priori that the generator has convex image — a restriction the present framework removes, since $f$5 is the natural domain of $f$6 regardless.

The equality problem — determining when two such means coincide for fixed $f$7 and fixed weights — is largely open even in the scalar case with discontinuous generators; recent work of Kiss and Pasteczka exhibits three-variable scalar QAMs that are equal without their generators being affine transforms of one another. The paper instead solves a particular instance: when does the vector-valued weighted QAM equal the weighted arithmetic mean? The answer, for the full-space domain $f$8, is clean and definitive:

Statement	Content
(i)	$f$9 on $f^{-1}$0
(ii)	$f^{-1}$1 with $f^{-1}$2 positive definite, $f^{-1}$3

The proof that (i) implies (ii) is the more delicate direction. Equality of the means forces $f^{-1}$4 to be injective, hence equal to the ordinary inverse; a supporting-hyperplane argument then shows $f^{-1}$5 has no boundary points in itself, so it is open. Brouwer's invariance of domain, applied to the continuous map $f^{-1}$6, yields continuity of $f^{-1}$7 itself. The two-variable instance of the mean equality reduces to a weighted Jensen equation $f^{-1}$8, whose coordinate-wise solutions are affine; positive definiteness of $f^{-1}$9 follows from strict monotonicity of $f^{(-1)}$0. The converse direction is a direct computation. This result is the vector-valued analogue of the classical Hardy–Littlewood–Pólya characterization in the scalar continuous case.

Limitations and open questions

The paper is explicit about the boundaries of its results. The characterization in the QAM theorem requires the domain to be all of $f^{(-1)}$1, and the author demonstrates by counterexample that this cannot be relaxed to arbitrary closed convex sets: on a degenerate domain $f^{(-1)}$2, any strictly increasing $f^{(-1)}$3 generates a QAM equal to the arithmetic mean, affine or not. However, this counterexample has empty interior, and the paper poses as an open problem whether the characterization holds for closed convex sets with nonempty interior. Two further open problems are stated: the equality problem for continuous generators on general domains $f^{(-1)}$4, and the extension of the entire theory to closed convex subsets of infinite-dimensional Hilbert spaces, where the finite-dimensional tools (Carathéodory's theorem, Helly's theorem, compactness of the sphere) are no longer available. It should also be noted that the continuity theorem and the existence theorem are proved only for closed convex domains, and the Okamura example indicates that no straightforward extension to open domains should be expected.

Conclusion

The paper resolves Páles's open problem affirmatively for closed convex domains: strictly monotone mappings on such sets possess a unique, explicitly constructible, monotone and continuous left-inverse on the convex hull of their image, with a global extension when the domain is compact. The counterexample based on the complex square map shows that closedness is indispensable, marking a genuine departure from the one-dimensional theory. The resulting theory of vector-valued weighted quasi-arithmetic means removes the convex-image assumption of prior work and yields a complete characterization of generators producing the arithmetic mean on the full space, while leaving the restricted-domain and infinite-dimensional generalizations open.