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It is said that the VC Dimension of a $d$-sided polygon is $2d+1(rectangle- 9)$, whereas there are proofs that the VC dimension of rotatable rectangle is 7. What is the reason for this anomaly?

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If I understand your question, it is not correct to say that the VC dimension of rectangles is $9$. For the class of rectangles with horizontal and vertical edges, the VC dimension is $4$ and for the class of rotated rectangles, the VC dimension is $7$. The class of $4$-sided polygon is a more wider class than the previous two. It does not require edges to be parallel nor the polygon to be convex. Therefore, it has more ability to shatter more vertices and thus has a greater VC dimension.

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