Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko Constructed Small Volume Bodies of Constant Width

Combinatorics and more 2024-06-01

From left to right: Andrii Arman, Andriy Bondarenko and Danylo Radchenko, Fedor Nazarov, and Andriy Primak.

The $n$ -dimensional unit Euclidean ball has width 2 in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them is 2.

There are other sets of constant width 2. A famous one is the Reuleaux triangle in the plane. The isoperimetric inequality implies that among all sets in $\mathbb R^d$ of constant width 2, the unit ball of radius 1 has maximum volume denoted by $V_n$ . (It is known that in the plane, the Reuleaux triangle has minimum volume.)

Oded Schramm asked in 1988 if for some $\varepsilon >0$ there exist sets $K_n$ , $n=1,2,\dots$ of constant width 1 in dimension $n$ whose volume satisfies

$\displaystyle \mathrm{vol}(K_n) \le (1-\varepsilon)^n V_n$ .

(See also this MO problem.)

Oded himself proved that the volume of sets of constant width 1 in $n$ dimensions is at least

$\displaystyle (\sqrt{3/2}-1+o(n))^n \cdot V_n$ .

In the paper Small volume bodies of constant width, Andrii Arman, Andriy Bondarenko, Fedor Nazarov, Andriy Prymak, and Danylo Radchenko, settled Schramm’s problem and constructed for sufficiently large $n$ a set $K_n$ of constant width 1 such that

$\displaystyle \mathrm{vol}(K_n) \le (0.9)^n V_n$ .

I am very happy to see this problem being solved. Congratulations Andrii, Andriy, Fedja, Andriy, and Danylo!

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