For a metric space $X$, the fixed points of $X$ are points that are fixed by all isometries of $X$. We will study computable properties of fixed points of compact subsets of $\mathbb{R}^n$. A natural question is: If $X$ is a computable subset of $\mathbb{R}^n$ whose set of fixed points is non-empty, is there a computable fixed point of $X$? The answer is negative and we will show an example of such $X$, but if the set of fixed points is finite, then the answer is positive. More generally, we will prove that the set of fixed points of a computable set is necessarily semicomputable. Additionally, we will show that the convexity of a computable set $X$ implies that the set of its fixed points is computable, so in that case $X$ contains computable fixed points.