## Background

The $n$-dimensional Euclidean ball of radius $1/2$ has width $1$ in every direction. Namely, when you consider a pair of parallel tangent hyperplanes in any direction the distance between them is $1$.

There are other sets of constant width $1$. A famous one is the Reuleaux triangle in the plane. The isoperimetric inequality implies that among all sets of constant width $1$ the ball has largest volume. Let's denote the volume of the $n$-ball of radius $1/2$ by $V_n$.

## The question

Is there some $\varepsilon >0$ so that for every $n>1$ there exist a set $K_n$ of constant width 1 in dimension n whose volume satisfies $\mathrm{vol}(K_n) \le (1-\varepsilon)^n V_n$.

This question was asked by Oded Schramm who also asked it for **spherical** sets of constant width r.

## A proposed construction

Here is a proposed construction (also by Schramm). It will be interesting to examine what is the asymptotic behavior of the volume. (And also what is the volume in small dimensions 3,4,...)

Start with $K_1=[-1/2,1/2]$. Given $K_n$ consider it embedded in the hyperplane of all points in $R^{n+1}$ whose $(n+1)$-th coordinate is zero.

Let $K^+_{n+1}$ be the set of all points $x$, with nonnegative $(n+1)$-th coordinate, such that the ball of radius $1$ with center at $x$ contains $K_n$.

Let $K^-_{n+1}$ be the set of all points $x$, with nonpositive $(n+1)$-th coordinate, such that $x$ belongs to the intersection of all balls of radius $1$ containing $K_n$.

Let $K_{n+1}$ be the union of these two sets $K^-_{n+1}$ and $K^+_{n+1}$.

## Motivation

Sets of constant width (other than the ball) are not lucky enough to serve as norms of Banach spaces and to attract the powerful Banach space theorist to study their asymptotic properties for large dimensions. But they are very exciting and this looks like a very basic question.

## References and additional motivation

In the paper: *"On the volume of sets having constant width"* Isr. J Math 63(1988) 178-182, Oded Schramm gives a lower bound on volumes of sets of constant width. Schramm wrote that a good way to present the volume of a set $K \subset R^n$ is to specify the radius of the ball having the same volume as $K$, called it the effective radius of the set $K$ and denote it by $\operatorname{er} K$. Next he defined $r_n$ as the minimal effective radius of all sets having constant width two in $R^n$. Schramm proved that $r_n \ge \sqrt {3+2/(n+1)}-1$. He asked if the limit of $r_n$ exists and if $r_n$ is a monotone decreasing sequence.

Our question is essentially whether $r_n$ tends to 1 as $n$ tends to infinity.

In the paper: O. Schramm, *Illuminating sets of constant width*. Mathematika 35 (1988), 180--189, Schramm proved a similar lower bound for the spherical case and deduced the best known upper bound for Borsuk's problem on covering sets with sets of smaller diameter.

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