Please excuse the vague title, but I fear the uncertainty of what to name it is at the heart of my question. Consider that I have a function which has some form of support in a (say 2-dimensional) space. I am particularly interested in how you call a support with the following properties: For any two points within the support, any point in-between is also supported.
Consider the following illustration: The support area is white, non-supported areas are red-dashed. Example a would not feature the desired properties, as the line between two points on opposite sides of its 'hole' fall outside the support. Similarly, two points (one located at the top, one at the bottom) at the left edge would also cross unsupported areas. Example b, on the other hand, would have the desired feature. So would, if I am not mistaken, a triangle, a rectangle, a pentagon, etc.
I considered for a while that the term I might be looking for could be compact support, but this only seems to mean that the support has finite limits, so if I understand the Wikipedia article correctly example a would also have compact support.
That sounds like the definition of a convex set.