Can We Determine If a Piecewise-Linear Surface Is Convex?

Title: Can We Determine If a Piecewise-Linear Surface Is Convex?

Research Question: Can we develop an efficient method to determine if a given piecewise-linear (PL) surface in three-dimensional space (or higher) is convex?

Methodology:

1. Define Local Convexity: We use Van Heijenoort's (1932) definition of local convexity. A surface is locally convex at a point if it has a neighborhood that lies on the boundary of a convex body.

2. Piecewise-Linear Surfaces: We consider PL surfaces, which are made up of simpler shapes called cells. Each cell is an affine subspace of the ambient space, and the surface is glued together from these cells.

3. Main Theorem: We prove that if a PL surface in three-dimensional (or higher) space has local convexity at every (n-3) cell, it is convex. This theorem provides a test for global convexity of PL surfaces.

Results:

1. Convexity Criterion: We develop a criterion for checking if a PL surface is convex. If local convexity holds at every (n-3) cell, the surface is convex.

2. Efficient Algorithm: We propose an efficient algorithm for checking local convexity at each (n-3) cell. The complexity of this algorithm depends on the way the surface is given as input data and the dimension of the space.

Implications:

1. Generalization of Van Heijenoort's Theorem: Our main theorem generalizes Van Heijenoort's theorem to spaces of constant curvature, including spherical and hyperbolic spaces.

2. Applications: Our method can be used in various fields that deal with PL surfaces, such as computer graphics, geometry, and topology. It can help in determining if a given surface can enclose a volume, which is useful in applications like packaging and architecture.

3. Further Research: Our method can be extended to other types of surfaces and spaces, and it can be combined with other algorithms to solve more complex problems.

Link to Article: https://arxiv.org/abs/0309041v1 Authors: arXiv ID: 0309041v1