Floating and Illumination Bodies for Polytopes: Duality results

Floating and illumination bodies for polytopes: duality results, Discrete Analysis 2019:11, 22 pp.

A well-known concept in convex geometry is that of the floating body of a convex body, which is defined as follows. If $\delta>0$ and $K$ is a (finite-dimensional) convex body, then $K_\delta$ is the set of all points $x\in K$ such that every hyperplane through $x$ cuts $K$ into two pieces of volume at least $\delta|K|$, where $|K|$ denotes the volume of $K$.

To understand why this is called the floating body (or the $\delta$-floating body if you want to make the parameter explicit), observe that the condition for a point $x\in K$ not to belong to $K_\delta$ is that there is some hyperplane that cuts off a piece of $K$ of volume less than $\delta$ that includes $x$. If we think of that hyperplane as being the surface of a lake, then if the convex body has an appropriate density relative to that of the water in the lake and is constrained not to rotate, then the part of $K$ cut off by the hyperplane is the part that is underwater. Thus the floating body $K_\delta$ consists of the parts of $K$ that will never go underwater if you rotate $K$ and gradually lower it into the lake. Note that $K_\delta$ is convex, since it is an intersection of half spaces: there are many results concerning relationships between properties of a convex body $K$ and properties of its floating body.

Another convex body that can be created out of a given body $K$ is its so-called illumination body. This one is a superset of $K$ rather than a subset. Again there is a parameter $\delta>0$, and $K^\delta$ is defined to be the set of all points $x$ such that the convex hull of $K\cup\{x\}$ has volume at most $(1+\delta)|K|$. If we think of $x$ as a light source, then $K\cup\{x\}\setminus K$ consists of all points that cast a shadow on the surface of $K$. It is not immediate that $K^\delta$ is convex, but this can be shown quite easily.

Of fundamental importance in convex geometry is a natural notion of duality. A hyperplane $H$ in $\mathbb R^n$ that does not contain 0 can be identified with the unique linear functional $\phi:\mathbb R^n\to\mathbb R$ such that $H=\{x:\phi(x)=1\}$. Then if $K$ is a convex body $K$ with 0 in its interior, its polar body $K^o$ consists of the set of all hyperplanes, interpreted as linear functionals (and therefore as elements of an $n$-dimensional vector space), that do not intersect the interior of $K$. It is easily shown that $K^{oo}=K$, so this is indeed a duality relation.

With this notion of duality, it is not hard to show that the operations of slicing off part of a convex body using a hyperplane, and taking the convex hull with a point, are dual in the following sense: if $K$ contains $0$ in its interior, $H$ is a hyperplane, and $\phi$ is the corresponding linear functional, then the polar of the convex body $K\cap\{x:\phi(x)\leq 1\}$ is the convex hull of $K^o\cup\{\phi\}$. This raises an obvious question: is the polar of the floating body equal to the illumination body of the polar?

The answer turns out to be no. As the authors remark, the illumination body of a polytope is always a polytope, and the polar of a polytope is a polytope, but floating bodies are always strictly convex, and in particular not polytopes. (This might seem to contradict the remarks in the previous paragraph, but it doesn’t, since $K^\delta$ is defined as a union of convex hulls, and unions do not dualize in a nice way.)

However, this observation leaves open the possibility of an approximate duality relation, which is potentially more interesting, since one would expect such a result, if true, to be true for deeper reasons than an exact relation.

This paper is one of two papers by the authors that investigate the extent to which the floating and illumination bodies are dual to each other. The companion paper to this one (Duality of floating and illumination bodies, arXiv:1709.02424) looks at the case of smooth bodies, whereas this one is concerned with polytopes. It is slightly surprising that the two cases are genuinely distinct, but they are: although polytopes are limits of smooth bodies and vice versa, the results are not “continuous” in the appropriate sense, and therefore different arguments are required.

The main result of this paper is a limiting result that states that an approximate duality relation holds when $\delta$ is very small (depending on the dimension). The proof involves significant technical arguments, but also a conceptual one, since the right formulation of the result is not obvious and relies on a new affine invariant of convex bodies that the authors introduce. Whether there is any kind of duality relation when $\delta$ is a fixed small constant remains an intriguing question.

This video, presented by Elisabeth Werner, one of the authors, gives an accessible introduction to floating bodies.