Timeline for When do the centers of mass of a uniform convex planar region as a whole and of its boundary alone coincide?

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when toggle format	what		by	license	comment
Jul 28, 2023 at 11:35	vote	accept	Nandakumar R
Jul 28, 2023 at 11:33	comment	added	Nandakumar R		Thanks again. Guess one can construct a convex planar region with both I and C coincident and without any symmetry by taking an equilateral triangle T and cutting off three small pieces with mutually different shapes from T's corners such that the resulting body T' is convex and also has same C as T. If the cut pieces are small, the I of T' will obviously be that of T. From your proof, T' will have both CMs coincident. Here, T' will have at least 6 sides. So finding a region with less sides (3 might not be possible) and with both CMs coincident might be more interesting.
Jul 28, 2023 at 10:20	comment	added	Karl Fabian		But note that the proof also uses that $P$ is star-shaped with respect to $I$.
Jul 28, 2023 at 10:13	comment	added	Karl Fabian		Thanks for the question. The proof uses the fact that by isotropic expansion around $I$ the faces move by a constant length, so touching is not reqired, but the incenter should be on the inside of all faces, i.e. in the halfspace in which the negative outer normal vector points.
Jul 28, 2023 at 4:26	comment	added	Nandakumar R		Thanks! Could you clarify if in the proof, the maximal inscribed sphere needs to touch all faces of the polyhedron? Iow, is the proof, by any chance, restricted to those polyhedrons wherein the max inscribed sphere touches all faces?
Jul 27, 2023 at 9:46	history	answered	Karl Fabian	CC BY-SA 4.0