[Paper Review] Hölder continuity of normal cycles and of support measures of convex bodies

Daniel Hug, Rolf Schneider|arXiv (Cornell University)|Oct 5, 2013

Point processes and geometric inequalities18 references3 citations

TL;DR

This paper establishes Hölder continuity estimates for the normal cycles and support measures of convex bodies in Euclidean space. Using the dual flat seminorm and bounded Lipschitz metric, it proves that the normal cycle varies Hölder continuously with exponent $1/(2n+1)$ in the Hausdorff metric, while support measures vary with exponent $1/2$, both with constants depending on geometric bounds of the bodies.

ABSTRACT

We provide an estimate of the distance (in the dual flat seminorm) of the normal cycles of convex bodies with given Hausdorff distance. We also give an estimate (in the bounded Lipschitz metric) of the support measures of convex bodies.

Motivation & Objective

• To provide a quantitative refinement of the known continuity of normal cycles of convex bodies under Hausdorff convergence.
• To establish local Hölder continuity of support measures with respect to the bounded Lipschitz metric.
• To improve upon existing stability results for curvature and area measures by providing explicit Hölder-type estimates.
• To extend techniques from Chazal et al. to convex bodies, enabling stronger estimates than previously available.
• To connect the regularity of normal cycles and support measures to geometric invariants such as inradius, circumradius, and parallel bodies.

Proposed method

• Estimate the dual flat seminorm of the difference of normal cycles using differential forms on $\mathbb{R}^{2n}$, leveraging smoothness and compact support on a common set $M$.
• Apply a modified version of the bounded Lipschitz metric to compare support measures $\Lambda_i(K,\cdot)$ and $\Lambda_i(L,\cdot)$, using the transformation formula and properties of distance and normal vector fields.
• Use the auxiliary measures $\mu_{K,\rho}$ defined via the projection and normal map on $K^\rho = K + \rho B^n$, which are shown to be Lipschitz continuous in the bounded Lipschitz metric.
• Decompose the difference in measures into three parts: overlap region with $|p_K - p_L|$, $|u_K - u_L|$, and symmetric difference $K^\rho \triangle L^\rho$, each estimated using geometric bounds.
• Apply known results on the gradient of distance functions ($\nabla d_K = u_K$) to bound the $L^1$-norm of normal vector differences.
• Solve a linear system with Vandermonde structure to express support measures $\Lambda_i$ as linear combinations of $\mu_{K,\rho_j}$ for $\rho_j = j/n$, enabling transfer of estimates.

Experimental results

Research questions

• RQ1 Can the weak continuity of normal cycles under Hausdorff convergence be strengthened to a quantitative Hölder estimate?
• RQ2 What is the optimal Hölder exponent for the normal cycle in the dual flat seminorm with respect to the Hausdorff metric?
• RQ3 How does the bounded Lipschitz distance between support measures of convex bodies scale with their Hausdorff distance?
• RQ4 Can the stability of Minkowski's existence and uniqueness theorem for area measures be improved via Hölder continuity of support measures?
• RQ5 To what extent can the methods of Chazal et al. be adapted to convex bodies to yield stronger regularity estimates?

Key findings

• The normal cycle $T_K$ satisfies the Hölder estimate $|T_K(\varphi) - T_L(\varphi)| \leq C(M,\varphi) \, d_H(K,L)^{1/(2n+1)}$ for all smooth $(n-1)$-forms $\varphi$ on a compact set $M$ containing $K_1 \times \mathbb{S}^{n-1}$ and $L_1 \times \mathbb{S}^{n-1}$.
• The support measures $\Lambda_i(K,\cdot)$ satisfy $d_{bL}(\Lambda_i(K,\cdot), \Lambda_i(L,\cdot)) \leq C(R) \, d_H(K,L)^{1/2}$, where $R$ bounds the parallel bodies $K_2$ and $L_2$.
• The area measure $S_{n-1}(K,\cdot)$ inherits the same Hölder estimate: $d_{bL}(S_{n-1}(K,\cdot), S_{n-1}(L,\cdot)) \leq C'(R) \, d_H(K,L)^{1/2}$.
• The constant $C(R)$ in the support measure estimate depends only on the dimension and the radius $R$ of a ball containing $K_2$ and $L_2$, not on the specific shape of $K$ and $L$.
• The proof relies on decomposing the bounded Lipschitz distance into three components: difference in position and normal vector fields on the intersection, and symmetric difference of parallel bodies, each bounded by $O(\sqrt{\delta})$.
• The linear reconstruction of $\Lambda_i$ from $\mu_{K,\rho_j}$ via Vandermonde systems ensures that the Hölder exponent of $1/2$ is preserved under linear combinations.