Special property of flat spaces

Question

I'm reading Eschenburg's paper Local convexity and nonnegative curvature — Gromov's proof of the sphere theorem recently. And I meet a little question. In 6.1 he said: let $M=\mathbb{R}^n$ be the euclidean $n$-space. Let $S$ be a connected hypersurface which is $\varepsilon$-convex with respect to the unit normal field $N$ on $S$. A special property of the flat space is $$ D^2 d(X, X) \geqq(d+\frac{1}{\varepsilon})^{-1}\|X\|^2 \text { for any } X \perp \nabla d $$ at any point where $d$ is smooth, also on $M_{+}^{\prime}$. I can't understand why this property holds on $M_{+}^{\prime}$, where $d>0$ and smooth.

Answer 1

I suppose that here $d$ is the signed distance to $S$. So the $\varepsilon$-convexity can be written as $$D^2d(X,X)\ge \varepsilon\cdot |X|^2$$ for any tangenet vector to $S$.

The level set $d=\ell$ is $\ell$-equidistant hypersurface, say $S_\ell$. So, Eschenburg says that in the Euclidean space, $S_\ell$ is $1/(\ell+\tfrac1\varepsilon)$-convex which is easy to check.