Theorem B — Finiteness of Doors

Prerequisites: Theorem A (energy isolation), Chapter 6 (doors).

Theorem B. For fruit $F$ with door threshold $\tau >0$:

(i) $|{\Sigma }_{\tau }(F,t)|\le \theta \cdot {\mathrm{vol}}_t(F)/\tau$.

(ii) ${\sum }_{p\in \Sigma }{e}_p\le \theta \cdot {\mathrm{vol}}_t(F)$.

(iii) $|{\Sigma }_{\tau }|/|F|\le \theta \cdot d_{\max }(F,t)/\tau$.

Proof. (Complete; reproduced from Chapter 6.)

(i) $\tau \cdot |{\Sigma }_{\tau }|\le {\sum }_{i\in {\Sigma }_{\tau }}{b}_{F,t}(i)\le {\sum }_{i\in F}{b}_{F,t}(i)={\mathrm{cut}}_t(F,\bar{F})\le \theta \cdot {\mathrm{vol}}_t(F)$.

(ii) ${\sum }_{p\in \Sigma }{e}_p={\sum }_{p\in \Sigma }{b}_{F,t}(p)\le {\mathrm{cut}}_t(F,\bar{F})\le \theta \cdot {\mathrm{vol}}_t(F)$.

(iii) ${\mathrm{vol}}_t(F)\le |F|\cdot d_{\max }(F,t)$. Substitute into (i). $\square$