Manuscript IX-D-ST@1 · IX-D-ST.tex · lemma · b1dc631a

Lemma: Sphere–ball intersection

The certificate and its lower-support consequence · lines 1884–1901 · proof included or linked

IX-D-ST@1#lem:sphereball

For $t > 0$ , $s \geq 0$ , and $R = ∣ x - y ∣$ ,

{∣ z - x ∣ = t} \cap {∣ z - y ∣ \leq s} \neq = \emptyset ⟺ t - s \leq R \leq t + s .

Proof. The minimum distance from $y$ to the sphere is $∣ R - t ∣$ , including $R = 0$ . The condition is $∣ R - t ∣ \leq s$ , which is equivalent to the displayed inequalities.

Displayed formulas in this passage

unlabeled · line 1888
${∣ z - x ∣ = t} \cap {∣ z - y ∣ \leq s} \neq = \emptyset ⟺ t - s \leq R \leq t + s .$

Nearby in this section

assumption: Assumption: Boundary-trace certificate (TC)
section: The certificate and its lower-support consequence
theorem: Theorem: Null-sphere lower support under (TC)
section: The certificate and its lower-support consequence
theorem: Theorem: The two-way bridge under (TC)
section: The certificate and its lower-support consequence
proposition: Proposition: Flip and conjugate transfer