凸体的膨胀同胚
Expensive Homeomorphism of Convex Bodies
AI总结：
本文利用Borsuk-Ulam定理和Brouwer不动点定理，通过归纳与边界延拓方法，证明了欧几里得空间中凸体内不存在膨胀同胚，揭示了凸性对映射的拓扑约束。
摘要：
本文针对欧几里得空间中凸体内是否存在膨胀同胚这一长期未决的问题展开研究。利用拓扑学的基本工具，包括Borsuk-Ulam定理和Brouwer不动点定理，我们证明了此类映射的不存在性。通过基于维数的归纳方法以及边界同胚的延拓，我们证明了膨胀同胚与这些体的紧致凸结构不相容。该工作揭示了拓扑原理与度量几何之间的相互作用，为凸性所施加的约束提供了新的见解。
英文摘要：
In this paper, we address the longstanding question of whether expansive homeomorphisms can exist within convex bodies in Euclidean spaces. Utilizing fundamental tools from topology, including the Borsuk-Ulam theorem and Brouwer's fixed-point theorem, we establish the nonexistence of such mappings. Through an inductive approach based on dimension and the extension of boundary homeomorphisms, we demonstrate that expansive homeomorphisms are incompatible with the compact and convex structure of these bodies. This work highlights the interplay between topological principles and metric geometry, offering new insights into the constraints imposed by convexity.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
