Dual Curvature Measures with Negative Indices

• Dual curvature measures with negative indices are defined via radial functions and the Gauss map, encoding essential geometric information for convex bodies.
• They guarantee existence and uniqueness for the dual Minkowski problem through variational methods and conditions on measure supports.
• Extensions to pseudo-cones and convex functions broaden the framework, linking the theory to analytic techniques, PDEs, and L_p formulations.

Dual curvature measures with negative indices comprise a central development in modern convex and discrete geometry, dual to Federer's curvature measures and closely interlinked with the dual Minkowski problem. For a convex body $K \subset \mathbb{R}^n$ containing the origin in its interior, the $q$-th dual curvature measure $\widetilde{C}_q(K,\cdot)$ is constructed via the radial function and the Gauss map, encoding surface-type geometric data. When the index $q < 0$, a full solution is available: necessary and sufficient conditions for prescribed measures, variational approaches, and robust uniqueness results. Extensions to pseudo-cones, general convex functions, and $L_p$ frameworks broaden the scope of dual curvature theory, revealing analytic, geometric, and PDE connections.

1. Foundational Definitions and Formulation

Let $K \subset \mathbb{R}^n$ be a convex body, $o \in \mathrm{int}\,K$, and consider:

• Radial function: $\rho_K(u) = \max\{\lambda > 0 : \lambda u \in K\}$, defined for $u \in S^{n-1}$.
• Support function: $h_K(v) = \max_{x \in K} x \cdot v$, defined for $v \in S^{n-1}$.

The $(n-q)$-th dual quermassintegral is

$$\widetilde{W}_{n-q}(K) = \frac{1}{n} \int_{S^{n-1}} \rho_K(u)^{q} \, du.$$

The normalized dual volume (for $q \neq 0$) is

$$\bar{V}_q(K) = \left( \frac{1}{n \omega_n} \int_{S^{n-1}} \rho_K(u)^{q} du \right)^{1/q},$$

with $\bar{V}_0(K)$ defined via the logarithmic mean.

For any Borel set $\eta \subset S^{n-1}$, Huang–Lutwak–Yang–Zhang define the $q$-th dual curvature measure as

$$\widetilde{C}_q(K, \eta) = \frac{1}{n} \int_{u \in \vec{\alpha}_K^*(\eta)} \rho_K(u)^{q} du,$$

where $\vec{\alpha}_K^*(\eta)$ collects directions whose corresponding boundary points have outer normals in $\eta$ (Zhao, 2017). Homogeneity holds: $$\widetilde{C}_q(\lambda K, \cdot) = \lambda^q \widetilde{C}_q(K, \cdot),$$ and $$\widetilde{C}_q(K, S^{n-1}) = \widetilde{W}_{n-q}(K)$$.

2. Minkowski-Type Existence and Uniqueness for $q < 0$

The dual Minkowski problem for $q < 0$ posits: Given a finite Borel measure $\mu$ on $S^{n-1}$ and $q < 0$, determine conditions for the existence of $K$ such that $\mu(\cdot) = \widetilde{C}_q(K, \cdot)$.

• Existence (Theorem A): $\mu$ must be a nonzero finite Borel measure not concentrated on any closed hemisphere. Then, there exists $K \in \mathcal{K}^n_0$ with $\mu = \widetilde{C}_q(K, \cdot)$.
• Uniqueness (Theorem B): If $K, L\in\mathcal{K}^n_0$, $q<0$, and $\widetilde{C}_q(K,\cdot)=\widetilde{C}_q(L,\cdot)$, then $K = L$ (Zhao, 2017).

The variational approach relies on a functional

$$\Phi(h) = -\frac{1}{|\mu|} \int_{S^{n-1}}\ln h(v)\,d\mu(v) + \ln \bar{V}_q([h]),$$

and coercivity for $q<0$ ensures compactness and existence, while the sign of $q$ yields robust uniqueness via contradiction arguments under rescaling.

3. Extensions: Pseudo-Cones and Convex Functions

a. Pseudo-Cones

Given a closed, pointed convex cone $C \subset \mathbb{R}^n$ with nonempty interior, a $C$-pseudo-cone is a closed convex set $K \subset C$, $o \notin K$, $\mathrm{rec}(K)=C$. The dual curvature measure with $q<0$ is

$$\widetilde{C}_q(K, \omega) = \frac{1}{n} \int_{\alpha_K^{-1}(\omega)} \rho_K(v)^q\,d\sigma(v),$$

for Borel sets $\omega \subset \mathrm{cl}\,\Omega_{C^\circ}$ ($\sigma$ spherical measure). The existence theorem states: Given any nonzero, finite Borel measure $\varphi$ supported in $\mathrm{cl}\,\Omega_{C^\circ}$ there exists a $C$-pseudo-cone $K$ with $\widetilde{C}_q(K, \cdot) = \varphi$ (Schneider, 10 Jan 2026). No compactness or interior-support conditions are required for $q<0$. Uniqueness remains open in this extension.

b. Convex Functions Framework

Dual curvature measures extend from convex bodies to proper convex functions $f:\mathbb{R}^n \to (0,\infty)$, via

$$\int_{\mathbb{R}^n} g(y) d\widetilde{C}_q(f,y) = \int_{\mathbb{R}^n} g(\nabla f(x)) f(x)^{1-q} e^{-|x|^2/2} dx.$$

The associated prescribed measure equation

$\mu = \tau\, \widetilde{C}_q(f, \cdot)$

admits solutions provided $\mu$ is finite, nonzero, not supported in a hyperplane, and $q < 0$ (Fang et al., 2021). This generalizes the convex body setting and introduces links to nonlinear PDE.

4. Analytical and Variational Structure

For $q<0$, the main analytical tools include variational functionals which are scale-invariant and log-concave under infimal convolution. The existence and uniqueness results follow from compactness in the space of convex bodies (or convex functions), first variation identities, and Minkowski-type inequalities for dual mixed quermassintegrals.

• In the pseudo-cone context, the functional

$$\Phi(f) = -\frac{1}{|\varphi|}\int\log f\,d\varphi + \frac{1}{q}\log\widetilde{V}_q([f])$$

is minimized on normalized dual volume level sets.

• For convex functions, the functional

$$\mathcal{V}_{\mu}(\varphi) = \int_{\mathbb{R}^n}\varphi(x)\,d\mu(x) - \exp\left(-W_{n+1-q}(\varphi^*)\right)$$

enables direct application of calculus of variations methodologies (Fang et al., 2021).

Negative indices $q<0$ guarantee coercivity, preclude degeneracy (shrinking/escaping bodies), and log-concavity crucial for variational solutions.

5. Geometric PDEs and Group Symmetry Considerations

The $L_p$ dual curvature density equations generalize the dual Minkowski problem to measures of the form $h_K^{p} \widetilde{C}_q(K,Q;\cdot)$. For $p<0$, convex bodies whose support functions solve

$\det(\nabla^2 h(u) + h(u)I) = f(u)\, h(u)^{p-1}$

on $S^{n-1}$ may be constructed under finite group symmetry assumptions, even beyond the origin-symmetric case. The existence theorem for such $L_p$ dual Minkowski problems holds for $-q^* < p < 0$ and $q>0$ (Böröczky et al., 13 Mar 2025). Regularity ($C^{2,\alpha}$) and strong convexity are established via elliptic estimates and group-invariant minimization.

Open questions remain regarding uniqueness, sharpness of critical exponents, Orlicz extensions, and extension of curvature flow approaches to $p<0$.

6. Connections to Classical Cases and Contrasts with $q > 0$

For $q=0$, the dual curvature measure coincides with Aleksandrov's problem, uniquely solved via optimal transport techniques. For $0 < q < n$, existence is solved for even measures, but uniqueness is an open problem except for $q=0$ (Zhao, 2017). The logarithmic Minkowski problem ($q = n$) is analogous to the cone volume measure and admits existence under symmetry or special measure conditions, but general uniqueness is unsolved.

In contrast, for $q < 0$, both existence and uniqueness are complete for all non-hemisphere-concentrated measures. For nonnegative $q$, analytic obstacles arise—loss of log-concavity, breakdown of Minkowski-type inequalities, and explicit counter-examples of non-existence due to mass concentration.

7. Illustrative Examples and Generalizations

For the uniform spherical measure, the unit ball $B^n$ solves the dual Minkowski problem for $q<0$. In the pseudo-cone setting, $K = C+z$ for $z \in \mathrm{int}\,C$ yields boundary-supported dual curvature measures.

Generalizations to functional, $L_p$, and Orlicz dual formulations allow prescription of more general measure densities and enable solution methods via PDE and advanced variational calculus (Fang et al., 2021, Böröczky et al., 13 Mar 2025).

The robust solution regime for negative indices in dual curvature measure theory demonstrates both deep analytic structure and broad geometric reach, underlying modern advances in convex geometric analysis.