Convex polygons in the plane $R^2$ arise in linear programming where the constraints are linear. The objective linear function attains its maximum at a vertex of the feasible region(if exists).
Assume that the constraints are presented by convex functions instead of linear ones. In this case, the feasible region if exists will be a region bounded by convex arcs(maybe line segments also). For example the conditions $y-x^2\le 5$ and $100x-(y-2)^2\le 100$ for non-negative values of $x,y$ gives such region as shown in the following figure.
enter image description here
My Question
Is there any study of such regions as generalized polygons considering its area, convexity, starshapedness, supporting lines and so on? I failed to find one using many keywords. So, if the answer is yes, please I need a reference and keywords if possible.
If the answer is no, do you think a study of these polygons is significant enough to consider as a research topic?
Thanks in advance.


OP: "please I need a reference and keywords if possible." There is literature that goes under the names "circular-arc polygons" and "conic-arc polygons."

Berberich, Eric, Arno Eigenwillig, Michael Hemmer, Susan Hert, Kurt Mehlhorn, and Elmar Schömer. "A computational basis for conic arcs and boolean operations on conic polygons." In European Symposium on Algorithms, pp. 174-186. Springer, Berlin, Heidelberg, 2002. Springer link
          Fig.1: The union of two curved polygons. Berberich et al.

Moore, Antoni, Chris Mason, Peter A. Whigham, and Michelle Thompson-Fawcett. "Polygon Generalization with Circle Arcs." In GSR. 2012. PDF download.

Wang, Zhi Jie, Xiao Lin, Bin Yao, Yong-Xi Gong, and Mei-E. Fang. "An Efficient Algorithm for Boolean Operation on Circular-arc Polygons." arXiv:1211.0729. (2012).

Aurenhammer, Franz, and Bert Jüttler. "On computing the convex hull of (piecewise) curved objects." Mathematics in computer science 6, no. 3 (2012): 261-266. Author PDF download.
          Aurenhammer & Jüttler.

Chernov, N., Yu Stoyan, Tatiana Romanova, and Aleksandr Pankratov. "Phi-functions for 2D objects formed by line segments and circular arcs." Advances in Operations Research 2012 (2012).
          Fig.1, Chernov, Romanova, Pankratov.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.