\ldots, R(x_k) + x) = f(x_1, \ldots, x_k)$ for all $x \in {\mathbf{R}}^d$, $R \in {\mathrm{O}}({\mathbf{R}}^d)$.

4.  $f \in C(U, [0, \infty[\;)$ is continuous; $f$ is always bounded from below by $0$; and if $k \geq 2$, $f$ is unbounded from above.

So $U$ and $f$ are invariant under all translations and orthogonal transformations in ${\mathbf{R}}^d$, but while $U$ is scaling-invariant, $f$ in general is not.

Let $\theta > 0$ and $(x_1, \ldots, x_k) \in U$. Then $$f(\theta \, x_1, \ldots, \theta \, x_k) = \frac{\alpha \, \theta^{\gamma}}{2} \sum_{j = 1}^l c_j \sum_{i \in e_j} \|x_i - z_j\|^{\gamma} + \frac{\beta}{2 \, \theta^{\delta}} \sum_{j = 1}^k \sum_{\substack{i = 1 \\ i \neq j}}^k w_i \, w_j \, \|x_i - x_j\|^{-\delta}.$$ So in particular, for all $(x_1, \ldots, x_k) \in U$ we have that $$\frac{1}{\beta} \left( \frac{\beta}{\alpha} \right)^{\frac{\delta}{\gamma + \delta}} \, f \left( \left( \frac{\beta}{\alpha} \right)^{\frac{1}{\gamma + \delta}} \, x_1, \ldots, \left( \frac{\beta}{\alpha} \right)^{\frac{1}{\gamma + \delta}} \, x_k \right) = \tilde{f}(x_1, \ldots, x_k),$$ where $\tilde{f}$ is given by , but with $\alpha = \beta = 1$. Therefore, we can pick $\alpha = \beta = 1$ without loss of generality; this just scales the minimum of $f$ by an overall factor (a similar 