Small volume bodies of constant width (2405.18501v2)

Abstract: For every large enough $n$, we explicitly construct a body of constant width $2$ that has volume less than $0.9^n \text{Vol}(\mathbb{B}^{n}$), where $\mathbb{B}^{n}$ is the unit ball in $\mathbb{R}^{n}$. This answers a question of O.~Schramm.

• The paper introduces a novel construction of a constant width 2 body in Rⁿ with volume under 0.9 times that of the unit ball, solving Schramm’s challenge.
• Its innovative combinatorial geometry methods demonstrate that the effective radius rₙ is less than 1 for all n ≥ 2, achieving rₙ < 0.9 in large dimensions.
• The findings resolve a longstanding geometric problem and provide new insights for applications in material science, engineering, and high-dimensional data analysis.

SMALL VOLUME BODIES OF CONSTANT WIDTH

The paper entitled "SMALL VOLUME BODIES OF CONSTANT WIDTH" by A. Arman, A. Bondarenko, F. Nazarov, A. Prymak, and D. Radchenko rigorously addresses a longstanding geometrical question initially proposed by O. Schramm. Specifically, it concerns the development of bodies of constant width with efficient volume characteristics within the n-dimensional Euclidean space, $R^n$. Notably, this research undertakes an explicit construction of bodies of constant width 2 that possess a notably smaller volume than the previously known bounds. This approach answers Schramm's inquiry regarding the existence of a body of constant width with an effective radius less than 1 for sufficiently large n.

Key Contributions

• Definition and Construction: A convex body K in $R^n$ is characterized by having a constant width w if the projection length in any line equals w. The associated concept is the effective radius, r, where the volume of K equals the volume of an r-sphere in $R^n$. The authors adeptly build upon Schramm's groundwork, providing a demonstrative construction for constant width bodies by establishing a body M with constant width 2, where the volume is less than 0.9 times the volume of the unit ball B.
• Theoretical Breakthrough: The authors affirmatively address the question expounded by Schramm regarding the non-trivial lower bound on effective radii and propose the existence of an ε > 0 such that the effective radius $r_n \leq 1 - ε$ for all $n \geq 2$. Through theoretical and constructive proofs, their work reveals that $r_n < 0.9$ for sufficiently large dimensions n.

Methodology

To construct the body M, the authors employ novel techniques within combinatorial geometry and intersection configurations of spheres. A significant methodological aspect involves the delineation of body M within the upper orthant in $R^n$. Claim 1 and Claim 2 rigorously show M's properties, ensuring constant width, while the volume estimations leverage geometric arrangements of orthants.

Numerical and Analytical Analysis

The paper provides detailed calculations for volume estimations through geometric algebra. Techniques include the evaluation of volumes across coordinate orthants and their subsequent aggregation under combinatorial configurations. Precise calculations showcase that the volume integral over a given orthant respects the upper volume limitations, crucially supporting the lower effective radius conjecture.

Implications and Future Research

The completion of this inquiry not only resolves Schramm's posed problem but also opens avenues for further exploration within differential and integral geometry. The findings suggest potential advancements in the understanding of geometric shape optimization, notably in fields concerning material sciences and architectural engineering where spatial efficiency from a volume perspective is pivotal.

Moreover, this research enriches the understanding of geometric properties in high-dimensional spaces, with possible implications for machine learning algorithm efficiency and complexity reduction. Future endeavors can further refine broader classes of constant width bodies and extend geometrical interpretations to non-Euclidean spaces.

In conclusion, the paper presents a comprehensive and thorough analysis that addresses a crucial question within geometric theory, advancing both theoretical and practical understanding of constant width bodies and their volumetric characteristics in high-dimensional geometries.