I am trying to understand a proof and it has the following property:
For any $\rho \in [0,1]$ and $\beta_1, \beta_2 \in [0,1]^n$, then one can write
$$\beta_1^{\rho}: = \rho \beta_1 + (1-\rho)\beta_2 $$
Is anyone able to understand the logic behind this?
Judging by the "$:=$" symbol, this is a definition of a new notation that they've just made up for use later in the text. So they're saying that from now on $\beta_1^{\rho}$ denotes the point defined in the right-hand side. And the expression on the right, when taken over all $\rho\in[0,1]$ represents the segment from $\beta_1$ to $\beta_2$.