An analytic solution to the Busemann-Petty problem on sections of convex bodies

We derive a formula connecting the derivatives of parallel section functions of an origin-symmetric star body in R^n with the Fourier transform of powers of the radial function of the body. A parallel section function (or (n-1)-dimensional X-ray) gives the ((n-1)-dimensional) volumes of all hyperplane sections of the body orthogonal to a given direction. This formula provides a new characterization of intersection bodies in R^n and leads to a unified analytic solution to the Busemann-Petty problem: Suppose that K and L are two origin-symmetric convex bodies in R^n such that the ((n-1)-dimensional) volume of each central hyperplane section of K is smaller than the volume of the corresponding section of L; is the (n-dimensional) volume of K smaller than the volume of L? In conjunction with earlier established connections between the Busemann-Petty problem, intersection bodies, and positive definite distributions, our formula shows that the answer to the problem depends on the behavior of the (n-2)-nd derivative of the parallel section functions. The affirmative answer to the Busemann-Petty problem for n\le 4 and the negative answer for n\ge 5 now follow from the fact that convexity controls the second derivatives, but does not control the derivatives of higher orders.Comment: 13 pages, published versio

Bust Portrait of Abraham Lincoln

Bust portrait of Abraham Lincoln taken by Alexander Gardner on 10 April 1865. [Ref: O-115.]https://scholarsjunction.msstate.edu/fvw-cdv/1269/thumbnail.jp

Lincoln and Tad

Photograph of Abraham Lincoln and Tad. The image is a variant of a photograph taken by Alexander Gardner on 10 April 1865, just four days before the president was assassinated. [Ref: Ostendorf, p. 220].https://scholarsjunction.msstate.edu/fvw-cdv/1307/thumbnail.jp

Gardner\u27s Sylvan Background Portrait of Abraham and Tad Lincoln

The carte de visite features a portrait of Abraham Lincoln and Tad Lincoln taken by Alexander Garner on 10 April 1865 in what would be Lincoln\u27s final photo session before his assassination. The sylvan background seen in this manifestation was added later by Gardner for artistic effect. [Ref: O-114.] The photograph features Abraham Lincoln seated at a table with his proper left elbow resting on the tabletop and a book in his proper left hand. Tad leans on the opposite side of the table with his proper forearm resting on the tabletop.https://scholarsjunction.msstate.edu/fvw-artifacts/1821/thumbnail.jp

Portrait of Seated Abraham Lincoln

Photograph of an engraving or lithograph of Alexander Gardner\u27s seated photograph of Abraham Lincoln. [Ref: O-79.]https://scholarsjunction.msstate.edu/fvw-artifacts/1875/thumbnail.jp

Gardner\u27s Sylvan Background Portrait of Abraham and Tad Lincoln

The carte de visite features a portrait of Abraham Lincoln and Tad Lincoln taken by Alexander Garner on 10 April 1865 in what would be Lincoln\u27s final photo session before his assassination. The sylvan background seen in this manifestation was added later by Gardner for artistic effect. [Ref: O-114.] The photograph features Abraham Lincoln seated at a table with his proper left elbow resting on the tabletop and a book in his proper left hand. Tad leans on the opposite side of the table with his proper forearm resting on the tabletop.https://scholarsjunction.msstate.edu/fvw-cdv/1597/thumbnail.jp

Portrait of Seated Abraham Lincoln

Photograph of an engraving or lithograph of Alexander Gardner\u27s seated photograph of Abraham Lincoln. [Ref: O-79.]https://scholarsjunction.msstate.edu/fvw-cdv/1294/thumbnail.jp

Brady\u27s Album Gallery. No. 605. Group of President Lincoln, Gen. McClellan, and Suite

This object is a mounted albumen print featuring a group photo of A. Lincoln, Ward Hill Lamon, and Army officers; The photograph was taken by Alexander Gardner at Antietam, MDhttps://scholarsjunction.msstate.edu/fvw-photographs/1001/thumbnail.jp

Brady\u27s Album Gallery. No. 605. Group of President Lincoln, Gen. McClellan, and Suite

This object is a mounted albumen print featuring a group photo of A. Lincoln, Ward Hill Lamon, and Army officers; The photograph was taken by Alexander Gardner at Antietam, MDhttps://scholarsjunction.msstate.edu/fvw-artifacts/1011/thumbnail.jp

11 x 11 Domineering is Solved: The first player wins

We have developed a program called MUDoS (Maastricht University Domineering Solver) that solves Domineering positions in a very efficient way. This enables the solution of known positions so far (up to the 10 x 10 board) much quicker (measured in number of investigated nodes). More importantly, it enables the solution of the 11 x 11 Domineering board, a board up till now far out of reach of previous Domineering solvers. The solution needed the investigation of 259,689,994,008 nodes, using almost half a year of computation time on a single simple desktop computer. The results show that under optimal play the first player wins the 11 x 11 Domineering game, irrespective if Vertical or Horizontal starts the game. In addition, several other boards hitherto unsolved were solved. Using the convention that Vertical starts, the 8 x 15, 11 x 9, 12 x 8, 12 x 15, 14 x 8, and 17 x 6 boards are all won by Vertical, whereas the 6 x 17, 8 x 12, 9 x 11, and 11 x 10 boards are all won by Horizontal