# Centroid within non-convex 2d polygon

The centroid of an object is defined as the arithmetic mean of all points of the object. For non-convex objects, the centroid is often not a part of the object itself:

Is there a definition of a centroid-like point which always lies within the object?

A definition that tackles 2 dimensional polygon objects is sufficient. I could think of something like the following, however, I would prefer a well-known definition if there is one:

1. Let $$s_p(a, b)$$ be the shortest path from $$a \in p$$ to $$b \in p$$ such that all points of the path are within $$p$$.

2. Let $$S_p := \{s_p(a, b) | a, b \in p\}$$.

3. Let $$d$$ be the longest path in $$S_p$$.

4. The mid point of $$d$$ is a centroid-like point and it always lies within $$p$$.

• But which properties should such a point have? You cannot expect it to retain all properties of the centroid. For instance: how would you choose this centroid-like point in the case of an annulus? Commented Jan 27, 2019 at 16:56
• The point should be some kind of central point that lies within the object. I dont know the exact properties because I am not that familiar with geometry. Are there any useful definitions of "special" points that always lie within the object? Commented Jan 27, 2019 at 18:38
• In the case of an annulus, for instance, any point you choose inside the object will not respect its rotational symmetry. Would that be fine for you? Commented Jan 27, 2019 at 19:15
• Yes, I think any point is fine in case of an annulus. However, if the annulus is sliced, I expect the point to be directly opposite to the slice. Maybe some mid point of some skeleton would do fine. Commented Jan 27, 2019 at 19:18