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Suppose we have a smooth curve $C$ lying in $\mathbb{R}^2$, and let us consider the orthogonal projection function $P_C(x)$ onto the curve, described by $$P_C(x) = argmin_{y \in C} \Vert x - y \Vert$$ where $\Vert \cdot \Vert$ is a norm, it can be $\Vert \cdot \Vert_2^2$, or $\Vert \cdot \Vert_1$.

My question is: is there a general relationship between the second derivative tensor of $P_C(x)$ and the curvature of the curve $C$? For example, relationship between their norms, and if the curve is convex, what can we say about the second derivative of the projection? If nothing specific can be said, under what restrictions on the curve $C$ and/or location of $x$ can we say something about their relationships? Does there exist work that discusses this problem or some problems related to it?

To visualize the problem somewhat, we consider an example here: Example

Denoting the blue curve as $C_1$ and black curve as $C_2$, $C_1$ clearly has greater curvature than $C_2$, but what about $\Vert D^2P_{C_1}(x) \Vert$ vs. $\Vert D^2P_{C_2}(x) \Vert$?

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