Editing Simplex (section)

=== Aitchison geometry ===
{{Main|Aitchison geometry}}

Aitchinson geometry is a natural way to construct an [[inner product space]] from the standard simplex <math>\Delta^{D-1}</math>. It defines the following operations on simplices and real numbers:

; Perturbation (addition)

:: <math> x \oplus y = \left[\frac{x_1 y_1}{\sum_{i=1}^D x_i y_i},\frac{x_2 y_2}{\sum_{i=1}^D x_i y_i}, \dots, \frac{x_D y_D}{\sum_{i=1}^D x_i y_i}\right]  \qquad \forall x, y \in \Delta^{D-1}
</math>

; Powering (scalar multiplication)

:: <math> \alpha \odot x = \left[\frac{x_1^\alpha}{\sum_{i=1}^D x_i^\alpha},\frac{x_2^\alpha}{\sum_{i=1}^D x_i^\alpha}, \ldots,\frac{x_D^\alpha}{\sum_{i=1}^D x_i^\alpha} \right]  \qquad \forall x \in \Delta^{D-1}, \; \alpha \in \mathbb{R}
</math>

; Inner product

:: <math> \langle x, y \rangle = \frac{1}{2D} 
\sum_{i=1}^D 
\sum_{j=1}^D
\log \frac{x_i}{x_j} 
\log \frac{y_i}{y_j} 
\qquad \forall x, y \in \Delta^{D-1}</math>