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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?

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    Sorry, when $\rho = 0$ we get that $1 = \beta_2$. Are you sure you didn't mean to write something like $\beta(\rho)= \rho\beta_1 + (1-\rho)\beta_2)$? – 2017-02-22
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    Nope. $\beta^{\rho}$. – 2017-02-22
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    It looks something like the projection of $\beta_{1}$ onto $\beta_{2}$. When $\rho=1$ it reproduces $\beta_{1}$ but when $\rho=0$ it gives $\beta_{2}$. – 2017-02-23

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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 $\beta_2$.