A lower bound for proportion of visibility polygon's surface to entire polygon's surface: Estimated by Art Gallery Problem and proven that cannot be greatly improved.

Assuming a bounded polygon and a point inside the polygon or on its boundary, the visibility polygon, also called the visibility region, is a polygon reachable, i.e., visible by straight lines from the point without hitting the polygon's edges or vertices. If the polygon is bounded, then the visibility polygon is bounded, and the proportion of the visibility polygon's surface area to the given polygon's surface area could be enumerated. Many papers investigate applications of the visibility polygons in robotics and computer graphics or focus on computationally effective finding the visibility region for a given polygon. However, surprisingly, there seems to be no work estimating the proportion of a visibility polygon's surface to an entire polygon's surface or its bounds. Thus, in this paper, we search for a lower bound of the surface proportion of a visibility polygon to a given one. Assuming $n$ -sided simple polygon, i.e., a polygon without holes and edge intersections, we apply the well-known art gallery problem and derive there is always a point inside the polygon or on its boundary that guarantees the proportion of the visibility polygon's surface to the entire polygon's surface is at least $\frac{1}{\lfloor n/3\rfloor}$ . We also show that there are $n$ -sided polygons for which the proportion of the visibility polygon's surface to the entire polygon's surface is asymptotically not greater than $\frac{1}{\lfloor n/3\rfloor}$ for any point inside the polygon or on its boundary. So, the lower bound of the proportion of the visibility polygon's surface to the entire polygon's surface, $\frac{1}{\lfloor n/3\rfloor}$ , cannot be improved in general.