Given a convex polyhedral surface P, we define a tailoring as excising from P a simple polygonal domain that contains one vertex v, and whose boundary can be sutured closed to a new convex polyhedron via Alexandrov's Gluing Theorem. In particular, a digon-tailoring cuts off from P a digon containing v, a subset of P bounded by two equal-length geodesic segments that share endpoints, and can then zip closed. In the first part of this monograph, we primarily study properties of the tailoring operation on convex polyhedra. We show that P can be reshaped to any polyhedral convex surface Q a subset of conv(P) by a sequence of tailorings. This investigation uncovered previously unexplored topics, including a notion of unfolding of Q onto P--cutting up Q into pieces pasted non-overlapping onto P. In the second part of this monograph, we study vertex-merging processes on convex polyhedra (each vertex-merge being in a sense the reverse of a digon-tailoring), creating embeddings of P into enlarged surfaces. We aim to produce non-overlapping polyhedral and planar unfoldings, which led us to develop an apparently new theory of convex sets, and of minimal length enclosing polygons, on convex polyhedra. All our theorem proofs are constructive, implying polynomial-time algorithms.

中文翻译：

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重塑凸多面体

给定一个凸多面体表面 P，我们将裁剪定义为从 P 中切除一个简单的多边形域，该域包含一个顶点 v，并且其边界可以通过亚历山德罗夫的胶合定理缝合到一个新的凸多面体上。特别地，一个 digon-tailoring 从 P 中切断一个包含 v 的 digon，一个 P 的子集，由共享端点的两个等长测地线段界定，然后可以压缩闭合。在本专着的第一部分，我们主要研究了凸多面体裁剪操作的性质。我们表明 P 可以通过一系列剪裁重新整形为任何多面体凸面 Q conv(P) 的子集。这项调查揭示了以前未探索的主题，包括将 Q 展开到 P 上的概念——将 Q 切割成不重叠地粘贴到 P 上的碎片。在本专着的第二部分，我们研究了凸多面体上的顶点合并过程（每个顶点合并在某种意义上都是 digon 裁剪的反向），将 P 嵌入到放大的表面中。我们的目标是产生不重叠的多面体和平面展开，这导致我们在凸多面体上开发了凸集和最小长度封闭多边形的明显新理论。我们所有的定理证明都是建设性的，暗示多项式时间算法。