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 Jul 13 awarded Commentator Jul 13 comment Integration problem in matrix calculus thx, this looks reasonable to me. Jul 13 accepted Integration problem in matrix calculus Jul 13 asked Integration problem in matrix calculus Nov 6 comment Efficient method for detecting a convex body in $\mathbb{R}^n$ Thanks @NielsDiepeveen for the remark. It is true that this algorithm can make a correct decision with high probability after testing enough pairs. I'm trying to figure out how high is that. Nov 5 awarded Scholar Nov 5 accepted Samples in the convex body vs. samples on the convex surface Nov 5 comment Samples in the convex body vs. samples on the convex surface Thanks @MikeSpivey for this nice and clear answer. The example is really convincing. Nov 5 awarded Supporter Nov 5 comment Efficient method for detecting a convex body in $\mathbb{R}^n$ Thanks @ChristianBlatter for the remark. It's true that a trivial sampler will take a long time to mix on such $K_1$. Fortunately, there is an algorithm for bringing a convex body into near-isotropic position. Nov 5 asked Samples in the convex body vs. samples on the convex surface Nov 5 revised Efficient method for detecting a convex body in $\mathbb{R}^n$ added 77 characters in body Nov 5 revised Efficient method for detecting a convex body in $\mathbb{R}^n$ deleted 8 characters in body Nov 5 comment Efficient method for detecting a convex body in $\mathbb{R}^n$ Thanks Christian Blatter, point taken. Nov 5 awarded Editor Nov 5 revised Efficient method for detecting a convex body in $\mathbb{R}^n$ added 358 characters in body Nov 5 comment Efficient method for detecting a convex body in $\mathbb{R}^n$ Thanks, but suppose one of these two bodies is a convex polytope, which means the boundary between $K_1$ and $K_2$ is a set of hyperplanes $\mathcal{H}=\{H_1,H_2,\ldots,H_n\}$. I think this approach will fail when two points are on the same hyperplane $H\in \mathcal{H}$. Nov 4 comment Efficient method for detecting a convex body in $\mathbb{R}^n$ Assume $K_0$ is a hypercube in $\mathbf{R}^n$. $K_1$ is a hypersphere in this cube and $K_2=K_1^c$. It satisfies the first assumption and $K_0$ is convex. Nov 4 comment Efficient method for detecting a convex body in $\mathbb{R}^n$ Besides three assumptions listed above, I don't know any other information about $K_1$ and $K_2$. The only information can be used is the membership oracle, which reports whether a query point $x$ is in $K_1$ or in $K_2$, but provides no other information. Nov 4 awarded Student