Weakly Mean-Convex Balls

• Weakly mean-convex balls are geometric objects defined by relaxed mean curvature conditions and characterized by optimal concavity properties.
• They underpin rigidity phenomena in potential theory, BV trace inequalities, and isoperimetric problems, ensuring that the ball is uniquely optimal under stability conditions.
• Their analysis extends to infinite-dimensional spaces and metric structures, offering insights into weak convexity, free boundary problems, and robust control applications.

Weakly mean-convex balls are geometric objects that arise in multiple analytic and geometric frameworks where the mean convexity condition—usually understood as the boundary having nonnegative mean curvature—is relaxed or imposed in a non-strict or distributional sense. They play a critical role in characterizing rigidity phenomena, optimality properties, and convexity stability in analysis, PDE, and geometric measure theory. The interplay between analytic concavity of relevant potentials and geometric shape rigidity is foundational to the theory and application of weakly mean-convex balls.

1. Analytic Foundations: Optimal Concavity and Potential Theory

Consider a convex domain $\Omega \subset \mathbb{R}^n$ and its $p$-capacitary potential $u$, solving

$\begin{cases} \operatorname{div}(|\nabla u|^{p-2}\nabla u) = 0 & \text{in %%%%3%%%%} \ u=1 & \text{on %%%%4%%%%} \ u\to 0 & \text{as %%%%5%%%%} \end{cases}$

The function $u$ is said to be $(1-p)/(n-p)$-concave if $u^{(1-p)/(n-p)}$ is convex in $\mathbb{R}^n\setminus\Omega$. That is, for $a = (1-p)/(n-p)$,

$u^a \text{ is convex}$

This “optimal power‐concavity” is attained if and only if $\Omega$ is a ball (Salani, 2012). More precisely, define $q = (n-p)/(p-1)$, so $a = -1/q$. For the ball $B_R(x_0)$,

$$u(x) = R^q / |x - x_0|^q, \qquad u^{-1/q}(x) = |x - x_0|/R$$

This linearity reflects the rigidity: such power-concavity can only happen for balls. By the Brunn–Minkowski inequality for $p$-capacity and its equality case, all level sets $\Omega(t) = \{ x: u(x) \geq t\}$ must be homothetic, forcing $\Omega$ to be a ball. This analytic mean-convexity translates into constant mean curvature of the boundary in a weak sense.

2. Geometric Rigidity, Trace Inequalities, and BV Functions

Weak mean-convexity plays a role in inequalities for functions of bounded variation (BV), particularly boundary trace inequalities: $$\|u - \operatorname{med}_{\partial\Omega} u\|_{L^1(\partial\Omega)} \leq C_{\operatorname{med}}(\Omega) |Du|(\Omega)$$

$$\|u - \overline{u}_{\partial\Omega}\|_{L^1(\partial\Omega)} \leq C_{\operatorname{mv}}(\Omega) |Du|(\Omega)$$

The optimal constants $C_{\operatorname{med}}, C_{\operatorname{mv}}$ are minimized precisely when $\Omega$ is a ball (Cianchi et al., 2013). For vanishing median, balls are unique minimizers in all dimensions; for vanishing mean, balls are minimizers in $n\geq3$, while in $n=2$, disks achieve the constant but “stadium-shaped” domains may also approximate the optimum, leading to the terminology “weakly mean-convex balls.” Here, small deformations of the boundary do not affect the optimal constant, and full geometric uniqueness fails—reflecting a weak form of mean-convexity.

3. Isoperimetric Problems Under Weighted Measures

In settings where volume and perimeter are weighted by radial densities $r^m, r^k$ (with $m,k>0$), rigidity occurs for balls provided certain stability conditions are met (Giosia et al., 2016). Specifically, balls are uniquely isoperimetric if

$0 < m \leq k - \frac{k}{n + k - 1}$

given also convexity of the generating curve. Stability—that the second variation is nonnegative—serves as a weak mean-convex condition. The limiting assumption is that the generalized mean curvature

$$H_{(f,g)} = \frac{g}{f} H + \frac{1}{f}\frac{\partial g}{\partial n}$$

is constant and the second variation is nonnegative. The classification is complete for $m \leq k-1$, with a small gap (the convexity of the generating curve) remaining for $m$ closer to the stability threshold.

4. Weakly $m$-Convex Sets: Topological and Combinatorial Properties

The concept extends to weakly $m$-convex sets, defined so that at each boundary point $x$ there exists an $m$-dimensional affine plane $L$ through $x$ not intersecting the set (Dakhil et al., 2017, Osipchuk, 2021). Closed sets are weakly $m$-convex if they are limits of open weakly $m$-convex sets; their combinatorial structure is nontrivial (e.g., a closed weakly $(n-1)$-convex set with nonempty set of $(n-1)$-nonconvexity points must have at least three components). These sets arise naturally in shadow problems and approximation theory, and their connectivity yields quantitative statements about geometric constructions.

5. Convexity Under Multifunctions and Nonlinear Mappings in Banach Spaces

For mappings $F = f + G$, with $G$ a closed convex multifunction metrically regular near $x_0$ and $f$ a $C^{1,1}$ mapping with sufficiently small Lipschitz constant, small balls $B(x_0,\varepsilon)$ are mapped to convex sets (Uderzo, 2015). This generalizes the Polyak convexity principle and yields “weakly mean-convex balls” in infinite-dimensional settings. The preservation of convexity under perturbations provides hidden geometric structure useful in set-valued optimization and robust control.

6. Weak Convexity in Spaces of Metric Structures and Outer Space

In Gromov–Hausdorff space, balls centered at the one-point space are convex in the weak sense: any two metric spaces in such a ball can be joined by a shortest curve lying entirely in the ball, but not all curves between such spaces remain within the ball (Klibus, 2018). Similar phenomena occur in “Outer space” (CV$_n$) equipped with the asymmetric Lipschitz metric, where “out-going balls” are weakly convex: for any two points $y,z$ in the ball, the balanced folding path between them remains in the ball, ensuring a weak form of convexity for these balls (Qing et al., 2017).

7. Rigidity From Lower Bounds on Mean Curvature

Sharp characterization results establish that if a convex body $K$ has its pointwise $k$-th mean curvature bounded below by

$$H_k(K,x) \geq \left[\frac{(n+1)\mathrm{Vol}(K)}{\mathcal{H}^n(\partial K)}\right]^k$$

almost everywhere, then $K$ must be a ball (Santilli, 2019). Such conditions allow the extension of rigidity to generalized mean-convex sets, even permitting singularities or only $\mathcal{C}^{1,1}$ regularity away from a small set. This is a powerful generalization: the lower bound renders “weak mean-convexity” sufficient for unique identification.

8. Applications and Broader Impact

Weakly mean-convex balls appear in shape optimization, PDE rigidity problems, spectral inequalities, and geometric flows. They furnish local and global minimizers of analytic quantities under non-strict mean-convexity, encode stability properties of nearly extremal geometric bodies (in mean width, $\ell$-norm, or isoperimetric inequalities), and are central to understanding foliation structures and free boundary minimal disks in convex domains (Haslhofer et al., 2023). In infinite-dimensional analysis, weak mean-convexity governs the existence/non-existence of balls maximizing probabilistic concentration ("most likely balls") in Banach spaces (Schmidt, 2023).

9. Summary Table: Key Characterizations

Context	Weak Mean-Convexity Criterion	Rigidity/Optimality Outcome
$p$-capacitary potential	$u^{(1-p)/(n-p)}$ convex	$\Omega$ must be a ball
BV trace inequalities	Ball minimizes $C_{\operatorname{med}}$, $C_{\operatorname{mv}}$	Strict/minimized/weak uniqueness
Isoperimetric weighted	Nonnegative second variation	Balls uniquely isoperimetric
Convex body curvature	$H_k(K,x)$ lower bound	Ball uniquely satisfies equality
Banach-space convexity	Multifunction metric regularity + $C^{1,1}$ perturbation	Image of small ball convex
Metric space (GH, Outer)	Weak convexity of balls (not strong)	Geodesics remain in ball (some)

10. Open Questions and Future Directions

The characterization of weakly mean-convex balls invites further paper of stability phenomena: how close must a nearly extremal convex body be to a ball? Removing technical assumptions (e.g., convexity of the generating curve in isoperimetric density problems) would fully resolve some conjectures. There are also open problems concerning minimal numbers of balls required for shadowing on spheres, classification of weakly $m$-convex sets, and extension to more general topologies and functional settings. Connections to free boundary regularity and geometric flows continue to develop, with implications for minimal surface theory, scalar curvature problems, and high-dimensional probability.

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Weakly mean-convex balls represent a fusion of analytic concavity, geometric rigidity, and topological stability across a spectrum of domains, revealing the structural singularity of the ball amidst relaxed convexity conditions and serving as a nexus for deep mathematical phenomena.