Directional Hadamard Differentiability

• Directional Hadamard differentiability is defined by uniform convergence in directions, offering sharp measure-theoretic and geometric insights.
• It characterizes non-differentiability sets using sigma-tangential and sigma–directionally porous concepts, crucial for robust analysis.
• The framework underpins transfer results from Gâteaux to Fréchet differentiability and has key applications in maximal operators and risk measures.

Directional Hadamard differentiability is a refined notion of differentiability for functions between infinite-dimensional spaces, Lipschitz mappings, and set-valued operators, which is especially relevant for nonsmooth analysis, geometric measure theory, and variational analysis. In the Hadamard sense, directional differentiability requires uniform convergence in direction (both in the base point and the direction variable), and often provides sharp, measure-theoretic and geometric information about the structure and size of the non-differentiability set. This framework underpins quantitative results for “almost everywhere” differentiability, measurable selection, and regularity transfer in functionals and maximal operators.

1. Definitions and Theoretical Foundations

Let $f: X \to Y$ be a mapping between Banach spaces, or more generally, between subsets of $\mathbb{R}^n$ or separable Banach spaces.

• Directional (Hadamard) Derivative: $f$ is said to be directionally Hadamard differentiable at $x \in X$ along $v \in X$ if, for every sequence $t_i \downarrow 0$ and $v_i \to v$, the limit

$$f_H(x,v) = \lim_{i \to \infty} \frac{f(x + t_i v_i) - f(x)}{t_i}$$

exists.

• Hadamard Differentiability: $f$ is Hadamard differentiable at $x$ if there exists a continuous linear map $L: X \to Y$ such that

$$\lim_{t \to 0} \sup_{v\in C} \left\| \frac{f(x+tv) - f(x)}{t} - L(v) \right\| = 0$$

for every compact $C \subset X$.

• One-sided Hadamard (Directional) Derivative: In non-symmetric contexts, one often considers

$$f_{H+}(x, v) := \lim_{z\to v,\, t\to0^+} \frac{f(x + t z) - f(x)}{t}.$$

The relation between Gâteaux differentiability (which only requires the existence of conventional directional derivatives) and Hadamard differentiability (which requires uniform convergence) is central. In locally Lipschitz or pointwise Lipschitz contexts, the two can coincide outside of small exceptional sets.

2. Geometric Structure of Non-differentiability Sets

For directionally differentiable Lipschitz functions, the structure of the non-differentiability set is governed by the geometry of the domain:

• $k$-Tangential Set: A set $E \subset \mathbb{R}^n$ is $k$-tangential if for every $x\in E$ there exists a $k$-dimensional linear space $V_x$ so that, for sequences $h_i\to0$ with $x+h_i \in E$, the transverse component $|h_i^{V_x^\perp}|/|h_i^{V_x}|\to0$.
• The set of points where the maximal differentiability degree of $f$ drops (i.e., the dimension along which linear approximation fails) can be decomposed as a countable union of $k$-tangential sets, denoted $\sigma$–$k$-tangential.
• Thus, the non-differentiability set is "slender" in a geometric sense: it is essentially contained in sets of small codimension; more precisely, for a directionally differentiable Lipschitz function, the failure of differentiability only happens on sets that are $\sigma$–tangential, and is thus negligible from the perspective of geometric measure theory (Luiro, 2012).

This structure is made quantitative using the metric

$$T(W, f, x) = \inf_{L \in \mathcal{L}(W)} \limsup_{w \to 0,\, w \in W} \frac{|f(x+w) - f(x) - L(w)|}{|w|}$$

which assesses linear approximability of $f$ restricted to $W$.

3. Exceptional Sets: Porosity, Nullity, and Smallness

The sets where Hadamard (or even Gâteaux) differentiability fails are captured by highly "thin" sets in the topological and measure-theoretical sense:

Set Type	Smallness/Null Property	Role in Differentiability
$\sigma$–directionally porous	Aronszajn null, Haar null, $\Gamma$-null, first category	Exceptions to Hadamard differentiability are always contained here (Zajicek, 2012, Zajicek, 2012)
$\sigma$–$k$-tangential	Conical, negligible in measure/geometric sense	Non-differentiability for directionally differentiable Lipschitz maps (Luiro, 2012)

Thus, even if $f$ is only Gâteaux differentiable or directionally differentiable in "many" directions, Hadamard differentiability (and sometimes even Fréchet differentiability in finite dimensions) holds except on a $\sigma$–directionally porous set.

4. Transfer, Extension, and Practical Use: From Directional Information to Full Differentiability

Key transfer results stipulate that:

• If $f$ is Gâteaux differentiable and Lipschitz at a point $x$, then $f$ is Hadamard differentiable at $x$, except on a $\sigma$–directionally porous set (Zajicek, 2012).
• If the one-sided Hadamard derivative $f_{H+}(x,u)$ exists for all $u$ in a dense set $S_x \subset X$, then $f$ is Hadamard differentiable at $x$ outside a $\sigma$–directionally porous set (Zajicek, 2012).
• In finite-dimensional spaces, Hadamard and Fréchet differentiability coincide, yielding almost everywhere (a.e.) Fréchet differentiability for everywhere Gâteaux differentiable functions (outside nowhere dense $\sigma$–porous sets).

These transfer principles are essential for analysis and optimization in Banach spaces and for extending Rademacher’s theorem to generalized contexts.

5. Applications to Maximal Operators and Functionals

The Hadamard directional differentiability framework is powerful for studying nonlinear, supremal, and maximal operators.

• Hardy–Littlewood Maximal Function: If $f$ is continuous and differentiable outside a $\sigma$–tangential set, and $Mf(x)$ is finite, then $Mf$ is also differentiable up to a $\sigma$–tangential set. In particular, if $f$ is differentiable a.e., then so is $Mf$ (Luiro, 2012).
• Supremum-type Functionals: The supremum, maximum norm, infimum, and amplitude functionals are all Hadamard directionally differentiable (but not fully Fréchet differentiable in infinite dimensions). The directional derivatives are computed via explicit formulas involving extremal points (see Theorem 2.1 of (Cárcamo et al., 2019)). These results enable functional delta-method theorems for the asymptotic analysis of statistics (e.g., Kolmogorov–Smirnov, Berk–Jones, MMD).
• Risk Measures and Statistical Applications: Risk functionals that are not classically differentiable can be handled via quasi-Hadamard or directional Hadamard differentiability, allowing accurate sensitivity and limit theorems in financial mathematics (Krätschmer et al., 2014).

6. Generalizations and Extensions: Infinite Dimensions, Manifolds, and Set-valued Analysis

The Hadamard directional differentiability concept admits several generalizations:

• Infinite-dimensional Banach Spaces: The structure of non-differentiability sets is preserved via porosity and $k$-tangentiality; Hadamard differentiability criteria depend on local or pointwise Lipschitzness and the denseness of the span of differential directions (Zajicek, 2012, Zajicek, 2013).
• Interval-valued and Manifold-valued Functions: On Hadamard manifolds or with interval values, the "directional" or generalized derivative may require a geodesic adaptation or nonstandard difference operations (e.g., generalized Hukuhara difference) (Nguyen et al., 2022, Bhat et al., 2022).
• Generalized Hadamard Differentiability: In empirical process theory and multivariate statistics, the concept is further relaxed to allow for small, asymptotically negligible perturbations, providing a robust foundation for weak convergence proofs (Neumeyer et al., 2023).
• Composite and Operator-level Analysis: For evolution operators, QVI solution operators, and sweeping processes, Hadamard directional differentiability yields linearized, optimality, and stationarity characterizations necessary for control and optimization, often under nonconvex and nonsmooth conditions (Alphonse et al., 2018, Alphonse et al., 2020, Christof et al., 2021, Brokate et al., 22 Mar 2025).

7. Further Directions and Open Problems

• Extending the theory to mappings that are not even pointwise Lipschitz, or to more general metric/Banach settings (possibly with non-separable target spaces), remains partly unresolved.
• Finer characterization of exceptional sets in infinite dimensions (e.g., precise relations between porosity and other nullness notions) is still developing.
• The connection with other generalized differentiability notions (Clarke, viscosity, codifferential, coexhauster) is the subject of ongoing integration (Abbasov, 2021, Jourani et al., 2021).
• Geometric characterizations (Clarke tangent cone containing a hyperplane, strict vs. directional Hadamard differentiability) offer criteria for linearization and regularity in variational analysis, optimization, and multiobjective problems (Jourani et al., 2021).

Directional Hadamard differentiability thus serves as the bridge between purely directional (Gâteaux-type) and fully uniform/strong differentiability (Fréchet), providing sharp geometric, measure-theoretic, and analytic tools essential for modern analysis in infinite dimensions, statistical modeling, optimal control, and beyond.