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Jaehong Oh

ℓPart 2· Theorem B

Theorem B — Finiteness of Doors

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

Theorem B. For fruit FF with door threshold τ>0\tau>0:

(i) ∣Στ(F,t)∣≤θ⋅volt(F)/τ|\Sigma_\tau(F,t)|\le\theta\cdot\mathrm{vol}_t(F)/\tau.

(ii) ∑p∈Σep≤θ⋅volt(F)\sum_{p\in\Sigma}e_p\le\theta\cdot\mathrm{vol}_t(F).

(iii) ∣Στ∣/∣F∣≤θ⋅dmax⁡(F,t)/τ|\Sigma_\tau|/|F|\le\theta\cdot d_{\max}(F,t)/\tau.

Proof. (Complete; reproduced from Chapter 6.)

(i) τ⋅∣Στ∣≤∑i∈ΣτbF,t(i)≤∑i∈FbF,t(i)=cutt(F,Fˉ)≤θ⋅volt(F)\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) ∑p∈Σep=∑p∈ΣbF,t(p)≤cutt(F,Fˉ)≤θ⋅volt(F)\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) volt(F)≤∣F∣⋅dmax⁡(F,t)\mathrm{vol}_t(F)\le|F|\cdot d_{\max}(F,t). Substitute into (i). □\square