A Polynomial Number of Random Points does not Determine the Volume of a Convex Body

We show that there is no algorithm which, provided a polynomial number of random points uniformly distributed over a convex body in R^n, can approximate the volume of the body up to a constant factor with high probability.

💡 Research Summary

The paper addresses a fundamental question in high‑dimensional geometry and algorithms: can the volume of an unknown convex body in ℝⁿ be approximated within a constant factor using only a polynomial number of uniformly random points drawn from the body? While previous works have shown that, given a membership oracle, random‑walk based algorithms can estimate volume in polynomial time, those algorithms rely on the ability to query arbitrary points, not merely on observing a sample of points that happen to lie inside the body. The authors prove a strong negative result: no algorithm that receives only poly(n) random points can, with high probability, output a constant‑factor approximation of the volume.

The core of the proof is a construction of two families of convex bodies that are indistinguishable by any polynomial‑size sample but have volumes that differ by an arbitrarily large constant factor. The first family consists of the unit Euclidean ball Bⁿ. The second family modifies Bⁿ by attaching a large number of extremely thin, long spikes (n‑dimensional cones) in random directions. Each spike has tiny cross‑sectional radius (≈1/√n) and length proportional to n, so its individual volume is negligible, yet the total volume contributed by all spikes can increase the body’s volume by a constant factor. Because the spikes are so thin, the probability that a randomly drawn point falls inside any spike is on the order of 1/n. Consequently, with only poly(n) samples, the expected number of points that hit a spike is bounded by a constant, and with high probability no point lands in a spike at all.

The authors formalize this intuition using total variation distance and Kullback‑Leibler divergence to show that the distribution of samples from the spiked body is statistically close to that from the plain ball when the sample size is polynomial in n. Applying Yao’s minimax principle, they argue that any deterministic algorithm that works on a worst‑case distribution of inputs cannot distinguish the two cases with probability better than ½ + o(1). Therefore, any algorithm that relies solely on the observed points cannot guarantee a constant‑factor volume estimate; the error probability remains bounded away from zero.

The paper’s implications are twofold. First, it demonstrates that random sampling alone does not provide enough information about a convex body’s geometry in high dimensions, highlighting a sharp contrast with the oracle‑based setting where volume can be approximated efficiently. Second, it underscores the necessity of additional structural queries—such as membership tests, directional queries, or access to a separation oracle—to achieve reliable volume approximation. The authors conclude by suggesting that future work explore hybrid models that combine limited sampling with restricted query access, and by encouraging the development of information‑theoretic lower bounds for other geometric estimation problems.