Here is a linear version of the Ehrenfest paradox with the goal of understanding the observations of someone in motion in the scenario, then solicit your views on whether the calculations are correct and whether one can extend it to circular motion. Consider a one dimensional train of proper length 100 traveling at 0.6c relative to a straight track. Gamma for the train relative to the ground is 1.25. The train's ground length is 80. The train is shunted onto a rectangular track whose short sides are so small as can be ignored, and whose long sides (side A and side B) each have 40 ground length. In the ground frame, the entire train fits in the rectangle with its front touching its rear (the train's ground length is 80, and the sum of sides A and B is 80). At any given time in the ground frame half of the train is moving to the right along side A at 0.6c, and half is moving to the left along side B at 0.6c. The train goes continuously around the rectangle. The train's proper length is not altered by going around the track -- it still has 100 proper length because each half of the train occupies 40 ground length of track traveling at 0.6c, so each half of the train still has 50 proper length. The relative speed of the sections of the train on the opposite sides of the rectangle is (by Einstein velocity addition) 0.8823529411764707, for which gamma equals 2.125. Although half of the train is on each side of the track in the ground frame, the same should not be true in the frame of an observer on the train because of the relativity of simultaneity. Any observer on the train is traveling inertially (except when he reaches the end of a side and turns around). In his inertial frame, both sides of the track are length contracted to length 32. Therefore the portion of the train's proper length that is in his frame (that is on the same section of track as he simultaneously in that frame) is 32. In his reference frame, the remaining 68 of the train's proper length is on the other side of the track. The 68 remaining proper length divided by 2.125 (the appropriate gamma) is 32. Therefore the total train length in his frame is 64. Therefore the train fits into the rectangular track in the ground frame because the 100 proper length is contracted by gamma=1.25 to a ground length of 80. The train fits into the rectangular track in the frame of an observer on the train because the track's length is contracted to 64 in his frame, and the train's length in his reference frame is also 64 (proper length divided by gamma squared). So one could say that the perimeter of the rectangle is 80 in the ground frame and 64 in the train observer's frame. But if the train observer walks around the train he would measure it to have proper length of 100 (it has not deformed in the length dimension), and the train is at all times within the rectangle, so he might also say that the perimeter of the rectangle is 100 for an observer on the train.