# 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? – Intelligenti pauca Jan 27 '19 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? – pschill Jan 27 '19 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? – Intelligenti pauca Jan 27 '19 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. – pschill Jan 27 '19 at 19:18