---
title: Dual Curvature Measures with Negative Indices
url: https://www.emergentmind.com/topics/dual-curvature-measures-with-negative-indices
type: topic
---

# Dual Curvature Measures with Negative Indices

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 $\eta$ [1703.00524]. 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 $K = L$ [1703.00524].

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 $\widetilde{C}_q(K, \cdot) = \varphi$ [2601.06594]. 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 $q < 0$ [2105.01250]. 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 [2105.01250].

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 $q>0$ [2503.10044]. 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 $q=0$ [1703.00524]. 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 [2105.01250, 2503.10044].

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.

Source: https://www.emergentmind.com/topics/dual-curvature-measures-with-negative-indices