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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 $$P_C(x) = \arg\min_{y \in C} \Vert x - y \Vert$$ 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 of $P_C(x)$ and the curvature of the curve $C$? For example, relationship between the norm, whether it is "positive definite", etc. If no, 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 the picture below: enter image description here 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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  • $\begingroup$ You mean the second derivative once you've chosen some coordinate chart on the curve? Otherwise, this doesn't make sense. And the answer to your question, I suspect, may depend on the chart. I haven't done any computations. $\endgroup$ – Ted Shifrin Dec 24 '18 at 2:12
  • $\begingroup$ @TedShifrin Yes once I have chosen some coordinate chart. Can you recommend any text/paper that discusses this problem or something related? $\endgroup$ – Longti Dec 24 '18 at 3:38
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Yes, there is. You can find it in Gilbarg and Trudinger in the appendix: Boundary curvatures and the Distance Function. The authors deal with the case in which you have an open set $\Omega\subset \mathbb{R}^n$ and the boundary $\partial\Omega$ is a manifold of class $C^2$. They characterize the second derivative of the distance function $d(x)=\text{dist}(x,\partial\Omega)$ at a point $x_0$ close to the boundary in terms of the principal curvatures of $\partial\Omega$ at $y_0\in\partial \Omega$ where $d(x_0)=\vert x_0-y_0\vert$. They prove that up to a change of coordinate axes you can write $$D^2 d(x_0)=\text{diag}[\frac{-\kappa_1}{1-\kappa_1d(x_0)},\cdots,\frac{-\kappa_{n-1}}{1-\kappa_{n-1}d(x_0)},0].$$ If you write the boundary $\partial\Omega$ near $y_0$ as the hypersurface $x_n=\varphi(x')$, where $x'=(x_1,\ldots,x_{n-1})$, then the principal curvatures $\kappa_i$ at $y_0$ are defined as the eigenvalues of $D^2_{x'} \varphi(y_0')$, where $y_0=(y_0',y_{0n})$. The change of coordinates is such that the $x_n$ coordinates axis lies in the direction of the normal to $\partial\Omega$ at $y_0$ and the other $x_i$ coordinates axes lie in the direction of the eigenvalues of $\kappa_i$.

If $n=2$ then $D^2_{x'} \varphi(y_0')=\varphi''(y_0')=\kappa_1$, which is proportional to the signed curvature to the curve $\frac{\varphi''(y_0')}{(1+(\varphi''(y_0'))^2)^{3/2}}$.

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