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The four color theorem states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color.

If all regions are convex (i.e. the region contains the whole line segment between any two distinct points of the region), is it possible to color the regions with no more than three colors?

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Not necessarily - consider any such map featuring a triangular region, where each of the three neighbouring regions borders each of the others. These four regions all touch each other so must all be different colours.

triangle with rays from each vertex to infinity

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    $\begingroup$ I hope you don't mind, but I added an image to illustrate your answer (currently waiting for approval). $\endgroup$
    – Paul Sinclair
    Commented Aug 31, 2021 at 19:09
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Three colours won't be enough for all maps.

Here's one troublesome map made of 6 rectangles. (EDIT: graphics cleaned up. Also, the rectangles are now squares.)

enter image description here

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