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

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

The -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 of constant width 2, the unit ball of radius 1 has maximum volume denoted by . (It is known that in the plane, the Reuleaux triangle has minimum volume.) 

Oded Schramm asked in 1988 if for some there exist sets ,   of constant width 2 in dimension whose volume satisfies

. 

(See also this MO problem.) 

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

.  

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  a set of constant width 1 such that

.

I am very happy to see this problem being solved. This new result may be the dawn of an asymptotic era in the study of sets of constant width. Congratulations Andrii, Andriy, Fedja, Andriy, and Danylo!