Self perimeter of convex sets

Abstract: This paper introduces a natural definition for the volume of the unit ball in $n$-dimensional normed spaces $\mathbb{R}^n$. This definition preserves the Euclidean relation $P(B)/V(B)=n$ between the perimiter and the volume of the unit ball $B$ in $R^n$. We show that this volume definition is invariant under origin-preserving affine transformations and polar duality. For $n=2$, we derive an explicit integral formula for the self-perimeter of the unit ball, extend it to non-centrally symmetric sets;. The construction is extended to $\mathbb{R}^n$ via a recursive integration over the boundary, utilizing $(n-1)$-dimensional volumes of planar intersections. Finally, we pose and discuss an Alexandrov-type problem for the associated surface measure, providing perturbative solutions in the 2D case. In particular we prove that, generically, any perturbation of the surface measure of the Euclidean 2-D disk yields a 4-fold symmetric convex set in the leading order.

• The paper introduces a recursive definition of volume and self-perimeter that ensures the Euclidean perimeter-to-volume ratio P(B)/V(B)=n for convex bodies.
• It employs boundary integration and affine invariance to analyze convex sets across dimensions, with explicit formulations in both planar and higher-dimensional cases.
• The work tackles an Alexandrov-type inverse problem, revealing symmetry-induced obstructions and opening new directions in convex and metric geometry.

Self-Perimeter and Volume in Convex Geometry: A Recursive Approach

Introduction

The paper "Self perimeter of convex sets" (2604.01950) undertakes a fundamental reconsideration of the perimeter and volume notions for convex bodies in finite-dimensional normed (Minkowski) spaces. Traditional frameworks—such as the Busemann and Holmes-Thompson definitions—prioritize geometric invariances or analytic properties but do not guarantee preservation of the Euclidean ratio between the perimeter (or surface area) and the volume of the unit ball, namely $P(B) / V(B) = n$ in dimension $n$. This work proposes and rigorously develops a natural and recursive definition of volume, tailored to enforce this Euclidean ratio intrinsically, and investigates its consequences in low and high dimensions, including the analysis of associated Alexandrov-type problems.

Theoretical Framework and Novel Definitions

The proposed approach introduces self-perimeter and self-volume as canonical measures for convex, centrally symmetric bodies $B \subset \mathbb{R}^n$. Employing a recursive scheme, the volume in dimension $n$ is constructed through boundary integration, using the self-volumes of $(n-1)$-dimensional slice intersections: $$V(B^{(k)}) = \frac{1}{k} \int_{\partial B^{(k)}} \frac{V(B^{(k-1)}(\nu_x))}{\mathcal{H}_{k-1}(B^{(k-1)}(\nu_x))} d\mathcal{H}_{k-1}(x),$$ where $\nu_x$ is the outer normal to the boundary, and $$B^{(k-1)}(\nu_x) = B^{(k)} \cap \Sigma^{k-1}(\nu_x)$$ is the $(k-1)$-dimensional section orthogonal to $\nu_x$. The one-dimensional base case sets $n$0.

For a convex body $n$1, this leads to the perimeter: $n$2 with the critical property that $n$3 holds for the unit ball $n$4 associated with the Minkowski norm.

This definition contrasts with Busemann's, which normalizes surface area integrals by the Euclidean measure of central cross-sections but does not enforce the Euclidean perimeter-to-volume ratio. The recursive self-volume construction is shown to be invariant under origin-preserving affine transformations and under polarity, essential for the internal consistency of the associated measure space.

Explicit Analysis in Two Dimensions

For planar ($n$5) sets, the self-perimeter attains an explicit formulation: $n$6 where $n$7 is the radial function. For centrally symmetric convex sets (CCS), the classic extremal values re-emerge: the self-perimeter ranges between 6 (regular hexagon) and 9 (triangle), with the Euclidean circle fixed at $n$8.

A key technical result asserts that the self-perimeter functional $n$9, for a convex set $B \subset \mathbb{R}^n$0 and reference point $B \subset \mathbb{R}^n$1, is strictly convex, admitting a unique minimizer—the "generalized centroid" [makeev2003]. This strict convexity is extended to higher dimensions for simplexes and product sets.

The paper elaborates on the modular periodicity of regular polygon self-perimeters, computing their explicit dependency on $B \subset \mathbb{R}^n$2 and connecting maximum/minimum behavior with the symmetry class, resonating with prior spectral results [ghandehari2019, martini2001].

Extension to Higher Dimensions

The recursive framework extends to all $B \subset \mathbb{R}^n$3, with perimeter and volume defined iteratively via sections and boundary integrals. Concrete calculations for the hypercube yield $B \subset \mathbb{R}^n$4, agreeing with the Euclidean volume and demonstrating the correctness of the construction. The comparison to Busemann's definition shows that, while the perimeter/volume ratio $B \subset \mathbb{R}^n$5 for the hypercube under Busemann's normalization diverges as $B \subset \mathbb{R}^n$6, the recursive self-volume maintains $B \subset \mathbb{R}^n$7 identically.

Further, the recursive procedure naturally decomposes over Cartesian products: for $B \subset \mathbb{R}^n$8 and $B \subset \mathbb{R}^n$9, $n$0. This multiplicativity, proven by induction, emphasizes the conceptual alignment with canonical measure product structures in convex and affine geometry.

Uniqueness of the optimal reference point is established for simplices—via an explicit barycentric-coordinate argument—which generalizes to product sets by structural recursion.

Alexandrov-Type Problem: Inverse Surface Measure Reconstruction

Building on this intrinsic perimeter/volume construction, the paper formulates an Alexandrov-type inverse problem: given a positive measure $n$1 on the sphere $n$2, does there exist a (centrally symmetric) convex body whose self-surface measure is exactly $n$3? For $n$4, the analysis is carried out via perturbations of the uniform measure and the Euclidean disk.

A nontrivial constraint emerges: only 4-fold symmetric (i.e., Fourier harmonics $n$5) perturbations of the density admit leading-order solvability for the radial perturbation—manifesting an explicit rigidity. The crucial leading-order asymptotic is

$n$6

subject to a nontrivial mean-value constraint. This notable structural restriction on surface measure inversion is closely related to rigidity phenomena in classic and discrete geometry [schneider2014convex].

Implications, Open Problems, and Future Directions

The new recursive definition restores a Euclidean proportionality law for perimeter and volume in arbitrary normed spaces, reconciling several classical pathologies inherent in Busemann- or Holmes-Thompson-based definitions for strongly non-Euclidean bodies. The invariance properties (affine, polarity) confirm the mathematical naturality and geometric consistency of the construction.

The inverse problem reveals spectral and symmetry-induced obstructions—even for infinitesimal perturbations—in reconstructing convex bodies from a prescribed self-surface measure, suggesting a rich structure for further exploration in convex, Finsler, and normed geometry.

The paper identifies open extremal questions for self-volume among centrally symmetric bodies in higher dimensions and conjectures convexity/minimization properties for arbitrary convex bodies beyond product and simplex forms.

Conclusion

By introducing a recursive, affine-invariant notion of self-volume and self-perimeter, this work shifts the focus from extrinsically normalized measures to those naturally encoding Euclidean geometric relations in normed spaces. The results provide crucial insight into the geometric and analytic structure of surface area, volume, and inverse surface measure problems, with immediate connections to classic extremal and inverse problems. These advances supply a rigorous basis for further exploration in convex geometry, isoperimetry, and metric measure theory, and suggest several new directions for research on intrinsic and functional geometric inequalities in general normed and Finsler spaces.