We consider the problem of computing the outer‐radii of point sets. In this problem, we are given integers $n, d$, and k, where $k \le d$, and a set P of n points in $\Re^d$. The goal is to compute the outer k‐radius of P, denoted by ${\cal R}_k(P)$, which is the minimum over all $(d-k)$‐dimensional flats F of $\max_{p \in P} d(p,F)$, where $d(p,F)$ is the Euclidean distance between the point p and flat F. Computing the radii of point sets is a fundamental problem in computational convexity with many significant applications. The problem admits a polynomial time algorithm when the dimension d is constant [U. Faigle, W. Kern, and M. Streng, Math. Program., 73 (1996), pp. 1–5]. Here we are interested in the general case in which the dimension d is not fixed and can be as large as n, where the problem becomes NP‐hard even for $k=1$. It is known that $R_k(P)$ can be approximated in polynomial time by a factor of $(1 + \varepsilon)$ for any $\varepsilon > 0$ when $d - k$ is a fixed constant [M. Bădoiu, S. Har‐Peled, and P. Indyk, in Proceedings of the ACM Symposium on the Theory of Computing, 2002; S. Har‐Peled and K. Varadarajan, in Proceedings of the ACM Symposium on Computing Geometry, 2002]. A polynomial time algorithm that guarantees a factor of $O(\sqrt{\log n})$ approximation for $R_1(P)$, the width of the point set P, is implied by the results of Nemirovski, Roos, and Terlaky [Math. Program., 86 (1999), pp. 463–473] and Nesterov [Handbook of Semidefinite Programming Theory, Algorithms, Kluwer Academic Publishers, Norwell, MA, 2000]. In this paper, we show that $R_k(P)$ can be approximated by a ratio of $O(\sqrt{\log n})$ for any $1 \leq k \leq d$, thus matching the previously best known ratio for approximating the special case $R_1 (P)$, the width of point set P. Our algorithm is based on semidefinite programming relaxation with a new mixed deterministic and randomized rounding procedure. We also prove an inapproximability result that gives evidence that our approximation algorithm is doing well for a large range of k. We show that there exists a constant $\delta > 0$ such that the following holds for any $0 < \eps < 1$: there is no polynomial time algorithm that approximates $R_k(P)$ within $(\log n)^{\delta}$ for all k such that $k \leq d - d^{\varepsilon}$ unless NP $\subseteq$ DTIME $[2^{(\log m)^{O(1)}}]$. Our inapproximability result for $R_k(P)$ extends a previously known hardness result of Brieden [Discrete Comput. Geom., 28 (2002), pp. 201–209] and is proved by modifying Brieden’s construction using basic ideas from probabilistically checkable proofs (PCP) theory.