Supporting quadratics and nearest-point normals

Programme proof text: AN-03 course-writing task, OpenAI Codex. CC0 1.0. Selection and prerequisite connections: GPT-6 Astra (OpenAI).

These are the complete quadratic-contact argument (S6–S7) and the nearest-point normal construction from Sections 1 and 7 of Curved weights and the directions in which support can end. The original notation and equation labels are retained. The associated differential-equation theorem is not needed here.

For a relatively closed set in an open , an exterior normal means and a real function , defined near , with and on the nearby part of . The notation below denotes all such pairs.

The used foundation proofs are supplied in the metric foundation, Sections 12–13, for compactness, minima, the fundamental theorem and the chain rule. When a kernel is smooth, Section “Taylor estimates on compact neighborhoods,” proves the full integral Taylor remainder for finite differentiability. For a twice continuously differentiable function and its quadratic Taylor polynomial at zero, subtracting the constant Hessian term from that proved formula gives

f(h)-f_2(h)=\int_0^1(1-s)\bigl(D^2f(sh)-D^2f(0)\bigr)[h,h]\,ds =o(|h|^2).

Indeed the absolute value divided by is at most half the maximum Hessian difference on the segment, tending to zero by continuity. This supplies the precise remainder in (S6). The full inverse and coordinate-completion proofs are Regular level sets and transverse convolution, Lemmas 0.1–0.2. The analytic pullback lesson, Lemma 3, additionally proves that the polynomial coordinate completion in the first argument has a real analytic local inverse. These exact earlier proofs supply the calculus used in the selection.

Quadratic contact with the same differential

Here is the local form of a support normal. Translate to zero. If witnesses , let be its quadratic Taylor polynomial there and choose any . On a sufficiently small neighborhood, $g(x)=f_2(x)-f(0)-\kappa|x|^2<0\quad(x\in F\setminus\{0\}),\qquad dg(0)=N. \tag{S6}$ Indeed , so after shrinking. Complete to smooth coordinates with last coordinate . The inverse-function theorem applies because . In these coordinates $F\cap\{y_n\geq0\}\subset\{0\}. \tag{S7}$ The transformation is smooth even though the original witness was only ; using the polynomial in (S6) avoids imposing regularity on that witness.

Nearest-point normals

For completeness, a nonempty proper relatively closed set in a connected open has an exterior normal at some boundary point. Choose , a closed ball , and points arbitrarily close to . For , choose a point of minimizing distance to ; it exists by compactness. Since , the distance is at most , and the minimizing point lies in . In a neighborhood of , the function $f(x)=-|x-y|^2$ has a maximum on at , and . Thus .