Suppose we have a known vector $X$ in $R^n$, and for any vector $Y$ in $R^n$, we impose on it the restriction that it must have a fixed correlation coefficient $r$ with $X$:
\begin{align*} \frac{\sum_{i=1}^{n} (X_i-\overline{X}) (Y_i-\overline{Y}) }{ \sqrt{ \sum_{i=1}^{n} (X_i - \overline{X} )^2 \sum_{i=1}^{n} (Y_i - \overline{Y} )^2 } } = r \end{align*}
For simplicity we can assume $\overline{X} = \overline{Y} = 0$:
\begin{align} \frac{ \sum_{i=1}^{n} X_i Y_i }{ \sqrt{ \sum_{i=1}^{n} X_i^2 \sum_{i=1}^{n} Y_i^2 } } = r \end{align}
If we fix $y_1, y_2, \cdots, y_{n-1}$ and let $y_n$ vary, then Eq. \ref{20130901corr} can be turned into a quadratic equation, and for it to have a solution, the discriminant must satisfy $\Delta \ge 0$, which gives us the following inequality:
\begin{align} \label{20130901delta} r^2 \sum_{i=1}^{n-1} X_i^2 \sum_{i=1}^{n-1} Y_i^2 -(1 - r^2) X_n^2 \sum_{i=1}^{n-1} Y_i^2 &\le (\sum_{i=1}^{n-1} X_i Y_i)^2 \end{align}
Notice that when $r^2 = 1$, this is just Cauchy-Schwarz inequality in the reverse direction, so the two sides must be equal, which can be true if and only if $Y = kX$.
When we allow more elements of $Y$ to vary, things gets a bit more complicated, but it still boils down to a second degree polynomial with multiple variables.
When $n=10$, and allowing three elements of $Y$ to vary, this can be visualized in a 3D Cartesian coordinate system in Mathematica:
SeedRandom[17]
x = RandomVariate[NormalDistribution[0, 1], 10]
y = RandomVariate[NormalDistribution[0, 1], 7]
expr = Re[Correlation[x, Join[y, {y1, y2, y3}]]]
ContourPlot3D[
expr == 0.1, {y1, -200, 200}, {y2, -300, 300}, {y3, -400, 400}]
ContourPlot3D[
expr == 0.3, {y1, -200, 200}, {y2, -300, 300}, {y3, -400, 400}]
ContourPlot3D[
expr == 0.5, {y1, -200, 200}, {y2, -300, 300}, {y3, -400, 400}]
ContourPlot3D[
expr == 0.7, {y1, -200, 200}, {y2, -300, 300}, {y3, -400, 400}]
The shape of the cap-shaped surface gets increasingly sharper as $r$ gets larger, making one wonder whether it will still be possible if $r$ gets very close to 1.