The Busemann-Petty problem (posed in 1956) has an interesting history. It asks the following question: if $K$ and $L$ are two origin-symmetric convex bodies in $\mathbb{R}^n$ such that the volume of each central hyperplane section of $K$ is less than the volume of the corresponding section of $L$:
$$\operatorname{Vol}_{n-1}(K\cap \xi^\perp)\le \operatorname{Vol}_{n-1}(L\cap \xi^\perp)\qquad\text{for all } \xi\in S^{n-1},$$
does it follow that the volume of $K$ is less than the volume of $L$: $\operatorname{Vol}_n(K)\le \operatorname{Vol}_n(L)?$

Many mathematician's gut reaction to the question is that the answer must be yes and Minkowski's uniqueness theorem provides some mathematical justification for such a belief---Minkwoski's uniqueness theorem implies that an origin-symmetric star body in $\mathbb{R}^n$ is completely determined by the volumes of its central hyperplane sections, so these volumes of central hyperplane sections *do* contain a vast amount of information about the bodies. It was widely believed that the answer to the Busemann-Problem must be true, even though it was still a largely unopened conjecture.

Nevertheless, in 1975 everyone was caught off-guard when Larman and Rogers produced a counter-example showing that the assertion is false in $n \ge 12$ dimensions. Their counter-example was quite complicated, but in 1986, Keith Ball proved that the maximum hyperplane section of the unit cube is $\sqrt{2}$ regardless of the dimension, and a consequence of this is that the centered unit cube and a centered ball of suitable radius provide a counter-example when $n \ge 10$. Some time later Giannopoulos and Bourgain (independently) gave counter-examples for $n\ge 7$, and then Papadimitrakis and Gardner (independently) gave counter-examples for $n=5,6$.

By 1992 only the three and four dimensional cases of the Busemann-Petty problem remained unsolved, since the problem is trivially true in two dimensions and by that point counter-examples had been found for all $n\ge 5$.
Around this time theory had been developed connecting the problem with the notion of an "intersection body". Lutwak proved that if the body with smaller sections is an intersection body then the conclusion of the Busemann-Petty problem follows. Later work by Grinberg, Rivin, Gardner, and Zhang strengthened the connection and established that the Busemann-Petty problem has an affirmative answer in $\mathbb{R}^n$ iff every origin-symmetric convex body in $\mathbb{R}^n$ is an intersection body. But the question of whether a body is an intersection body is closely related to the positivity of the inverse spherical Radon transform. In 1994, Richard Gardner used geometric methods to invert the spherical Radon transform in three dimensions in such a way to prove that the problem has an *affirmative* answer in three dimensions (which was surprising since all of the results up to that point had been *negative*). Then in 1994, Gaoyong Zhang published a paper (in the Annals of Mathematics) which claimed to prove that the unit cube in $\mathbb{R}^4$ is not an intersection body and as a consequence that the problem has a negative answer in $n=4$.

For three years everyone believed the problem had been solved, but in 1997 Alexander Koldobsky (who was working on completely different problems) provided a new Fourier analytic approach to convex bodies and in particular established a very convenient Fourier analytic characterization of intersection bodies. Using his new characterization he showed that the unit cube in $\mathbb{R}^4$ *is* an intersection body, contradicting Zhang's earlier claim. It turned out that Zhang's paper was incorrect and this re-opened the Busemann-Petty problem again.

After learning that Koldobsky's results contradicted his claims, Zhang quickly proved that in fact every origin-symmetric convex body in $\mathbb{R}^4$ *is* an intersection body and hence that the Busemann-Petty problem has an affirmative answer in $\mathbb{R}^4$---the opposite of what he had previously claimed. This later paper was also published in the Annals, and so Zhang may be perhaps the only person to have published in such a prestigious journal both that $P$ and that $\neg P$!

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