Let $x_1, \ldots, x_N \in \mathbb{R}^D$ each be a D4924: Euclidean real number.
A D4924: Euclidean real number $a \in \mathbb{R}^D$ is an arithmetic mean of $x_1, \ldots, x_N$ if and only if \begin{equation} \sum_{n = 1}^N a = \sum_{n = 1}^N x_n \end{equation}

Let $x_1, \ldots, x_N \in \mathbb{R}^D$ each be a D4924: Euclidean real number.
A D4924: Euclidean real number $a \in \mathbb{R}^D$ is an arithmetic mean of $x_1, \ldots, x_N$ if and only if \begin{equation} \underbrace{a + a + \cdots + a}_{N \text{ times}} = x_1 + x_2 + \cdots + x_N \end{equation}