Commit bd4b5fb6 by Conor McCoid

### Tetra: dropped extraneous 0 in Gamma case from lemma on intersection value in higher dimensions

parent 852076a9
 ... ... @@ -1019,13 +1019,9 @@ Suppose we are $m$ steps through this algorithm and we are considering the inter \begin{lemma} Suppose the $m$--face of $X$ between the set of $m+1$ vertices $\Set{\vec{x}_i}{i \in J}$ intersects the $(n-m)$--hyperplane defined as the intersection of the $m$ hyperplanes $\Set{P_\gamma}{\gamma \in \Gamma}$. Denote this intersection as $\vec{h}(J | \Gamma)$. If $0 \notin \Gamma$ then % now that 1-x = x * e0 the separation of cases is completely unnecessary Then \begin{equation*} \vec{h}(J | \Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_\eta & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ \vec{x}_{i_m} \cdot \vec{e}_\eta & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}{\begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}} \end{equation*} and if $0 \in \Gamma$ then \begin{equation*} \vec{h}(J | \Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_\eta & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_0} \cdot \vec{e}_\gamma \\ \vdots & \vdots & & \vdots \\ \vec{x}_{i_m} \cdot \vec{e}_\eta & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_m} \cdot \vec{e}_\gamma \end{vmatrix}}{\begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_0} \cdot \vec{e}_\gamma \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & 1 - \sum_{\gamma \notin \Gamma} \vec{x}_{i_m} \cdot \vec{e}_\gamma \end{vmatrix}}. \vec{h}(J | \Gamma) \cdot \vec{e}_\eta = \frac{\begin{vmatrix} \vec{x}_{i_0} \cdot \vec{e}_\eta & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ \vec{x}_{i_m} \cdot \vec{e}_\eta & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}{\begin{vmatrix} 1 & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_0} \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{i_m} \cdot \vec{e}_{\gamma_m} \end{vmatrix}}. \end{equation*} \end{lemma} ... ... @@ -1038,7 +1034,7 @@ The $m$--face can be defined by $\Set{\vec{g}(\set{a_i})}{0 \leq a_i \leq 1}$, w \end{align*} The intersection $\vec{h}(J | \Gamma) = \vec{g}(A)$ depends on $\Gamma$. If $\Gamma$ does not contain 0 then we seek the set $A = \set{a_1, \dots, a_m}$ such that We seek the set $A = \set{a_1, \dots, a_m}$ such that \begin{align*} \vec{g}(A) \cdot \vec{e}_\gamma = 0 \ \forall \ \gamma \in \Gamma. \end{align*} ... ... @@ -1067,20 +1063,6 @@ The constant $d$ is found by rearranging the formula for $1-\sum a_i$: \implies d = & \begin{vmatrix} 1 & \vec{x}_1 \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_1 \cdot \vec{e}_{\gamma_m} \\ \vdots & \vdots & & \vdots \\ 1 & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_m} \end{vmatrix}. \end{align*} As a small validation of this formula, if $\vec{x}_i \cdot \vec{e}_\eta = c$ then $\vec{g}(A) \cdot \vec{e}_\eta = c$. If $\Gamma$ contains 0 then we seek the set $A$ such that \begin{align*} \vec{g}(A) \cdot \vec{e}_\gamma = & 0 \ \forall \ \gamma \in \Gamma \setminus \set{0}, \\ \sum_{\gamma \notin \Gamma} \vec{g}(A) \cdot \vec{e}_\gamma = & 1. \end{align*} The first of these conditions ensures that the first $m$ columns of the determinants within each $a_i$ remains unchanged. The last column of each is no longer necessary and is replaced by some unknown vector. To find the unknown vector we use the last condition listed above on $\vec{g}(A)$ and the established value of $d$: \begin{align*} 0 = & 1 - \sum_{\gamma \notin \Gamma} \vec{g}(A) \cdot \vec{e}_\gamma \\ = & d - \begin{vmatrix} \sum_{\gamma \notin \Gamma} \vec{x}_1 \cdot \vec{e}_\gamma & \vec{x}_1 \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_1 \cdot \vec{e}_{\gamma_{m-1}} & w_1 \\ \vdots & \vdots & & \vdots & \vdots \\ \sum_{\gamma \notin \Gamma} \vec{x}_{m+1} \cdot \vec{e}_\gamma & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_{m-1}} & w_{m+1} \end{vmatrix} \\ = & \begin{vmatrix} 1-\sum_{\gamma \notin \Gamma} \vec{x}_1 \cdot \vec{e}_\gamma & \vec{x}_1 \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_1 \cdot \vec{e}_{\gamma_{m-1}} & w_1 \\ \vdots & \vdots & & \vdots & \vdots \\ 1-\sum_{\gamma \notin \Gamma} \vec{x}_{m+1} \cdot \vec{e}_\gamma & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_1} & \dots & \vec{x}_{m+1} \cdot \vec{e}_{\gamma_{m-1}} & w_{m+1} \end{vmatrix}. \end{align*} Therefore, $w_i = 1 - \sum \vec{x}_i \cdot \vec{e}_\gamma$. \end{proof} Note that the $(n-m)$--face of $Y$ on the intersection of $\Set{P_\gamma}{\gamma \in \Gamma}$ is on the positive side of the remaining planes $\Set{P_\eta}{\eta \notin \Gamma}$. ... ...
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