Best and random approximation
of a convex body by a polytope

A common and important question in mathematics is whether mathematical objects or constructions with certain requirements or features exist. Such questions naturally appear in various areas of mathematics and theoretical computer science. For instance, in numerical linear algebra, where randomization is used for designing fast algorithms [40], in algorithmic convex geometry, in relation to the complexity of volume computation of high-dimensional convex bodies [2, 65], in geometric functional analysis, when one is interested in constructing normed spaces with certain pathological characteristics, or in graph theory, where one is interested in finding graphs with certain pre-described features. We will elaborate on the latter two and further examples below, where the corresponding references will be provided.

With a view towards motivation, let us stick with the last topic in that list for a moment, graph theory. The chromatic number of a graph is the least amount of colors necessary to color the vertices such that no two adjacent vertices share the same color and the girth of a graph is the length of a shortest cycle contained in the graph. In 1959, Erdős proved that there are graphs whose girth and chromatic numbers are both arbitrarily large [19]. Another classical result in graph theory is the following: there is a constant $c\in(0,\infty)$ such that for all sufficiently large $n\in{\mathds{N}}$ there exists a graph with $n$ vertices which contains no triangle and which does not contain a set of $c\sqrt{n}\ln n$ independent vertices; recall that a set of independent vertices in a graph is a set of vertices such that no two vertices represent an edge of the graph. In this case, Erdős could not give an explicit construction, but he showed that such graphs exist with high probability [20].

Erdős’ probabilistic approach is commonly referred to as the probabilistic method today and he was arguably the most famous of its pioneers (although others before him proved theorems using this method). Also Shannon employed this method in the proof of his famous source coding theorem [63]. The probabilistic method has become mathematical folklore way beyond graph theory to use it in order to show the existence of objects with prescribed features.

Let us continue our motivation on more analytic grounds. A famous theorem of Dvoretzky [17] states that in every infinite dimensional Banach space there are finite dimensional subspaces of arbitrarily large dimension that are up to a small error Euclidean spaces. More specifically, let $X$ and $Y$ be two normed spaces. The Banach-Mazur distance between $X$ and $Y$ is

$$d(X,Y):=\inf\big{\{}\|A\|\|A^{-1}\|\,:\,A:X\to Y\hskip 5.69054pt\mbox{is an isomorphism}\big{\}},$$

and, in case the spaces are not isomorphic, $d(X,Y):=+\infty$. Dvoretzky’s theorem says: there is a constant $c\in(0,\infty)$ such that for every $\varepsilon>0$ and every finite dimensional normed space $X$ there is a subspace $H$ of $X$ such that

$$\operatorname{dim}(H)\geq c\varepsilon^{2}\ln(\operatorname{dim}(X))\hskip 28.45274pt\mbox{and}\hskip 28.45274ptd(H,\ell_{2}^{\operatorname{dim}(H)})\leq 1+\varepsilon,$$

where $\ell_{2}^{k}$ denotes the Euclidean space of dimension $k$. Also this theorem is proved by the probabilistic method. But even more is proved: not only there exists such a subspace, but actually most of the subspaces satisfy these specifications. This leads to an interesting paradoxon: whenever one tries to select a subspace that is almost an Euclidean subspace one fails, but if one chooses a subspace randomly, then with high probability one chooses a subspace that is almost Euclidean.

As it turns out, Dvoretzky’s theorem is related to a fundamental problem in Quantum Information Theory. The goal is to determine the capacity of a quantum channel to transmit classical information and a question that naturally arose in this context, and which has been one of the major open problems for more than a decade, asks about the additivity of the so-called $\chi$-quantity for any pair of quantum channels. A more tractable, but equivalent question (see [64]), concerns the additivity of the minimal output von Neumann entropy of quantum channels. Eventually, this reformulation allowed Hastings [33] to construct his famous counterexample. Shortly after, it was discovered by Aubrun, Szarek, and Werner in [4, 5] that the existence of this counterexample is a consequence of Dvoretzky’s theorem.

A classical result in convex geometry and the local theory of Banach spaces is John’s theorem [38] (see also [6]) on maximal volume ellipsoids in convex bodies, which shows that, for any $n$-dimensional normed space $X$,

$$d(X,\ell_{2}^{n})\leq\sqrt{n}.$$

The latter implies that $d(X,Y)\leq n$ for any pair $X$, $Y$ of $n$-dimensional normed spaces. It might come as a surprise that $d(\ell_{1}^{n},\ell_{\infty}^{n})$ is only of order $\sqrt{n}$ and this raises the question whether or not a pair of spaces can be constructed that has Banach-Mazur distance of order $n$. The following major achievement is due to Gluskin [23]: there is a constant $c\in(0,\infty)$ such that for all $n\in\mathbb{N}$ there are $n$-dimensional normed spaces $X$ and $Y$ such that

$$d(X,Y)\geq c\,n.$$

Again, this result is proved by the probabilistic method and involves certain symmetric random polytopes acting as unit balls of the random normed spaces.

Finally, let us mention some recent results from information-based complexity (IBC) concerning the quality of random information in approximation problems compared to optimal information. A typical question in IBC is to approximate the solution of a linear problem based on $n\in{\mathds{N}}$ pieces of information about the unknown problem instance. Usually, it is assumed that some kind of oracle is available which grants us this information at our request and we may call this oracle $n$ times to get $n$ well-chosen pieces of information, trying to obtain optimal information about the problem instance. However, often this model is too idealistic and there might be no oracle at our disposal and the information just comes in randomly. In [35], the authors studied the circumradius of the intersection of an $m$-dimensional ellipsoid $\mathcal{E}$ with semi-axes $\sigma_{1}\geq\dots\geq\sigma_{m}$ with random subspaces of codimension $n$, where $n$ can be much smaller than $m$, and proved that under certain assumptions on $\sigma$, this random radius $\mathcal{R}_{n}=\mathcal{R}_{n}(\sigma)$ is of the same order as the minimal such radius $\sigma_{n+1}$ with high probability. In other situations $\mathcal{R}_{n}$ is close to the maximum $\sigma_{1}$. The random variable $\mathcal{R}_{n}$ naturally corresponds to the worst-case error of the best algorithm based on random information for $L_{2}$-approximation of functions from a compactly embedded Hilbert space $H$ with unit ball $\mathcal{E}$. In particular, $\sigma_{k}$ is the $k$th largest singular value of the embedding $H\hookrightarrow L_{2}$. In this formulation, one may also consider the case $m=\infty$ and it was shown that random information underlies an $\ell_{2}$-dichotomy in that it behaves differently depending on whether $\sigma\in\ell_{2}$ or not. We also refer the reader to subsequent works [36, 41] and the survey [34].

In this paper we give an overview of some results concerning best and random approximation of convex bodies by polytopes and how both are linked. As we shall see random approximation is almost as good as best approximation.

The accuracy of approximation of a convex body by a polytope is interesting in itself, but it is also relevant in many applications, for instance in computervision ([55], [54]), tomography [21], geometric algorithms [18].

We shall briefly set out the notation and some basic concepts used in this paper. By $B_{2}^{n}(x,r)$ we denote the closed Euclidean ball with center $x\in{\mathds{R}}^{n}$ and radius $r\in(0,\infty)$. The Euclidean norm on $\mathbb{R}^{n}$ is denoted by $\|\ \|_{2}$ and we write $S^{n-1}:=\{x\in{\mathds{R}}^{n}\,:\,\|x\|_{2}=1\}$ for the Euclidean unit sphere. The standard inner product on ${\mathds{R}}^{n}$ is denote by $\langle\cdot,\cdot\rangle$. A convex body $K$ in $\mathbb{R}^{n}$ is a compact convex set with non-empty interior. For such a body, the surface measure on its boundary, we write $\mu_{\partial K}$, is the restriction of the $(n-1)$-dimensional Hausdorff measure $\mathcal{H}^{n-1}$ to $\partial K$. For $x\in\partial K$ the normal at x to $\partial K$ is denoted by $N_{K}(x)$. $N_{K}(x)$ is almost everywhere unique. We shall denote by $\kappa_{K}(x)$ the Gauß-Kronecker curvature of $\partial K$ at $x$.

The Hausdorff distance between two convex bodies $C$ and $K$ in $\mathbb{R}^{n}$ is

$$d_{H}(C,K):=\inf\{t>0\,:\,C\subseteq K+tB_{2}^{n}\hskip 2.84526pt\mbox{and}\hskip 2.84526ptK\subseteq C+tB_{2}^{n}\}.$$

The symmetric difference distance or Nikodym metric is defined as

$$d_{S}(C,K):=\operatorname{vol}_{n}(C\triangle K)=\operatorname{vol}_{n}(C\setminus K)+\operatorname{vol}_{n}(K\setminus C)=\operatorname{vol}_{n}(C\cup K)-\operatorname{vol}_{n}(C\cap K),$$

where $\operatorname{vol}_{n}(\cdot)$ refers to the $n$-dimensional Lebesgue measure. We focus in this paper on the symmetric difference metric.

For points $x_{1},\dots x_{N}\in\mathbb{R}^{n}$ we denote by

$$[x_{1},\dots x_{N}]=\left\{\lambda_{1}x_{1}+\dots+\lambda_{N}x_{N}\,:\,\forall 1\leq i\leq N:0\leq\lambda_{i}\leq 1\hskip 2.84526pt\mbox{and}\hskip 2.84526pt\sum_{i=1}^{N}\lambda_{i}=1\right\}$$

the convex hull of these points. For two sequences $(a_{n})_{n\in{\mathds{N}}},(b_{n})_{n\in{\mathds{N}}}\in{\mathds{R}}^{\mathds{N}}$, we write $a_{n}\sim b_{n}$ if $\lim_{n\to\infty}a_{n}/b_{n}=1$. Moreover, if the sequences are non-negative, then we use the notation $a_{n}\lesssim b_{n}$ to indicate that there exists a constant $C\in(0,\infty)$ such that, for all $n\in{\mathds{N}}$, $a_{n}\leq Cb_{n}$. We write $a_{n}\approx b_{n}$ if both $a_{n}\lesssim b_{n}$ and $b_{n}\lesssim a_{n}$.

Bronshteĭn and Ivanov [13] and Dudley [16] proved independently that for every convex body $C$ in $\mathbb{R}^{n}$ there is a constant $a_{C}:=a_{C}(n)\in(0,\infty)$ such that for all $N\in\mathbb{N}$ there exists a polytope $P_{N}$ with at most $N$ vertices such that

$$d_{H}(C,P_{N})\leq\frac{a_{C}}{N^{\frac{2}{n-1}}},$$

(3.1)

i.e., with respect to the Hausdorff distance, convex bodies can be approximated arbitrarily well by polytopes (see also [57]). Note that the dependence in $N$ best possible. Since

$$d_{S}(C,K)\leq d_{H}(C,K)\,\mathcal{H}_{n-1}(\partial(C\cup K)),$$

the inequality carries over to the symmetric difference metric. Nearly two decades later, the estimate in (3.1) has been made more precise by Gordon, Meyer, and Reisner [24], showing the following: there is a constant $c\in(0,\infty)$ such that for all convex bodies $K$ in $\mathbb{R}^{n}$ and all $N\in\mathbb{N}$ there is a polytope $P_{N}$ having at most $N$ vertices such that

$$d_{S}(K,P_{N})\leq\frac{c\cdot n}{N^{\frac{2}{n-1}}}\operatorname{vol}_{n}(K),$$

(3.2)

i.e., the constant $a_{c}$ has been specified. Of course, it remains to understand the constant $c$ and also whether the dependence in terms of volume of the body $K$ is optimal or if this dependence can be replaced by a quantity that is related to the surface structure of $K$. We will come back to this question later.

Macbeath [44] observed that for every convex body $C$ in $\mathbb{R}^{n}$ with $\operatorname{vol}_{n}(C)=\operatorname{vol}_{n}(B_{2}^{n})$, for every $N\in\mathbb{N}$ and every polytope $P_{N}$ contained in $B_{2}^{n}$ having at most $N$ vertices there is a polytope $Q_{N}$ contained in $C$ having at most $N$ vertices and

$$\operatorname{vol}_{n}(P_{N})\leq\operatorname{vol}_{n}(Q_{N}),\hskip 28.45274pt\mbox{i.e.}\hskip 28.45274ptd_{S}(Q_{N},C)\leq d_{S}(P_{N},B_{2}^{n}).$$

Therefore, to prove the inequality (3.2) it is enough to show it for the Euclidean ball (and polytopes contained in the Euclidean ball). By a result of Kabatjanskii and Levenstein [39] for any given angle $\phi$ there are $\xi_{1},\dots,\xi_{N}\in S^{n-1}$ such that

	$\displaystyle\cos\phi\geq\langle\xi_{i},\xi_{j}\rangle\hskip 56.9055pti\neq j$
	$\displaystyle\forall x\in S^{n-1}\exists i:\hskip 28.45274pt\cos\phi\leq\langle x,\xi_{i}\rangle$
	$\displaystyle N\leq(1-\cos\phi)^{-\frac{n-1}{2}}2^{0.901(n-1)}.$

Choosing $P_{N}=[\xi_{1},\dots,\xi_{N}]$, we obtain

$$\operatorname{vol}_{n}(B_{2}^{n})-\operatorname{vol}_{n}(P_{N})\leq 2^{1.802-2}\frac{n\cdot\operatorname{vol}_{n}(B_{2}^{n})}{N^{\frac{2}{n-1}}}.$$

(3.3)

On the other hand, Gordon, Reisner and Schütt [25, 26] showed the following.

Here the condition $N\geq(b\cdot n)^{\frac{n-1}{2}}$ is needed to ensure the specific constant in (3.4). If we drop the assumption $N\geq(b\cdot n)^{\frac{n-1}{2}}$, then the estimate becomes

$$\frac{a\cdot\operatorname{vol}_{n}(B_{2}^{n})}{N^{\frac{2}{n-1}}}\leq\operatorname{vol}_{n}(B_{2}^{n})-\operatorname{vol}_{n}(P_{N}).$$

We believe though that inequality (3.4) holds without this assumption.

Let us see if more precise estimates for best approximation of a convex body by polytopes can be obtained. The following result relates the symmetric difference between a convex body with sufficiently smooth boundary and a polytope contained in it with the so-called Delone triangulation constant and the affine surface area of $K$.

The integral expression

$$\int_{\partial K}\kappa_{K}(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)$$

is a fundamental quantity in convex geometry and called the affine surface area of $K$.

Theorem 3.2 was proved by McClure and Vitale [47] in dimension 2 and by Gruber [31, 32] in higher dimensions.

By (3.3) and (3.4) it follows that the constant del_n-1 is of the order of $n$, which means that there are numerical constants $a,b\in(0,\infty)$ such that, for all $n\in\mathbb{N}$,

$$a\cdot n\leq\mbox{del}_{n-1}\leq b\cdot n.$$

(3.5)

In two papers by Mankiewicz and Schütt the constant del_n-1 has been better estimated [45, 46]. It was shown there that

$$\frac{\tfrac{n-1}{n+1}}{\operatorname{vol}_{n-1}(B_{2}^{n-1})^{\frac{2}{n-1}}}\leq\operatorname{del}_{n-1}\leq(1+\tfrac{c\ln n}{n})\frac{\tfrac{n-1}{n+1}}{\operatorname{vol}_{n-1}(B_{2}^{n-1})^{\frac{2}{n-1}}},$$

(3.6)

where $c\in(0,\infty)$ is a numerical constant. It follows from Stirling’s formula that

$$\lim_{n\to\infty}\frac{\mbox{del}_{n-1}}{n}=\frac{1}{2\pi e}=0.0585498....$$

The right-hand inequality of (3.6) is proved using a result by Müller [48] on random polytopes of the Euclidean ball. It is a special case of Theorem 4.2. Altogether we have quite precise estimates for best approximation of convex bodies by polytopes if the numbers of vertices of the polytopes are big.

We try now to give a more precise estimate for best approximation for the whole range of the numbers $N$ of vertices. While the asymptotic estimate for large $N$ involves the affine surface area, we involve in general the convex floating body. We explain why this is quite natural and, of course, what the convex floating body is.

Let $K$ be a convex body in $\mathbb{R}^{n}$ and $t\geq 0$. The convex floating body $K_{t}$ of $K$ is the intersection of all halfspaces $H^{+}$ whose defining hyperplanes $H$ cut off a set of volume $\delta$ from $K$, i.e.,

$$K_{\delta}=\bigcap_{\operatorname{vol}_{n}(H^{-}\cap K)=\delta}H^{+}.$$

Let us introduce the notion of generalized Gauß-Kronecker curvature. A convex function $f:X\rightarrow\mathbb{R},X\subseteq\mathbb{R}^{d}$ is called twice differentiable at $x_{0}$ in a generalized sense if there are a linear map $d^{2}f(x_{0})\in L(\mathbb{R}^{d})$ and a neighborhood $U(x_{0})$ so that we have for all $x\in U(x_{0})$ and all subdifferentials df(x)

$$\parallel df(x)-df(x_{0})-(d^{2}f(x_{0}))(x-x_{0})\parallel_{2}\leq\Theta(\parallel x-x_{0}\parallel_{2})\parallel x-x_{0}\parallel_{2},$$

where $\lim_{t\to 0}\Theta(t)=\Theta(0)=0$ and where $\Theta$ is a montone function. $d^{2}f(x_{0})$ is symmetric and positive semidefinite. If f(0)=0 and df(0)=0 then the ellipsoid or elliptical cylinder

$$x^{t}d^{2}f(0)x=1$$

is called the indicatrix of Dupin at 0. The general case is reduced to the case f(0)=0 and df(0)=0 by an affine transform. The eigenvalues of $d^{2}f(0)$ are called the principal curvatures and their product the Gauß-Kronecker curvature $\kappa(0)$.

Busemann and Feller [14] proved for convex functions mapping from $\mathbb{R}^{2}$ to $\mathbb{R}$ that they have almost everywhere a generalized Gauss-Kronecker curvature. Aleksandrov [1] extended their result to higher dimensions.

The expression

$$\int_{\partial K}\kappa_{K}(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)$$

is called affine surface area of $K$. It is generally conjectured that Theorem 3.2 holds for arbitrary convex bodies if we substitute the classical Gauß-Kronecker curvature by the generalized one.

Since the affine surface area is an integral part of the formula for best approximation in the case of a large number of vertices and since the affine surface area is given by (3.7) one can speculate whether in the case of general numbers $N$ of vertices the floating body is involved. In fact, this is the case.

Let us describe the floating body algorithm. We are choosing the vertices $x_{1},\dots,x_{N}\in\partial K$ of the polytope $P_{N}$. $x_{1}$ is chosen arbitrarily. Having chosen $x_{1},\dots,x_{k}$, we choose a support hyperplane $H_{k+1}$ to $K_{\delta}$ such that

$$K_{\delta}\subseteq H_{k+1}^{+}\hskip 28.45274pt\mbox{and}\hskip 28.45274ptx_{1},\dots,x_{k}\in H_{k+1}^{+}.$$

If there is no such hyperplane the algorithm stops. We choose $x_{k+1}\in\partial K\cap H_{k+1}^{-}$ such that the distance of $x_{k+1}$ to $H_{k+1}$ is maximal. This means that $x_{k+1}$ is contained in the support hyperplane $H$ to $K$ that is parallel to $H_{k+1}$ and such that $H\subseteq H_{k+1}^{-}$.

We want to see how good this algorithm is. Therefore, we determine what Theorem 3.4 gives in the case of a smooth body and in the case of a polytope. First the case of a smooth body.

The main difference between the optimal result and (3.9) is that $n^{2}$ appears as a factor on the right-hand side of (3.9) and not $n$. It still an open question whether or not an improvement of the argument gives the same estimate but with $n$ instead of $n^{2}$.

On the other end of the spectrum of convex bodies are the polytopes. Of course, best approximation of a polytope by polytopes is a trivial task once the number of vertices one may choose is equal to or greater than the number of vertices of the polytope. We cannot expect that Theorem 3.4 gives us this. But, what does Theorem 3.4 give in the case of a polytope?

Let us denote the set of all polytopes in $\mathbb{R}^{n}$ by $\mathcal{P}^{n}$. The extreme points or vertices of a polytope $P$ shall be denoted by $\operatorname{ext}(P)$.

A $n+1$-tuple $(f_{0},\dots,f_{n})$ such that $f_{i}$ is an $i$-dimensional face of a polytope $P$ for all $i=0,1,\dots,n$ and such that

$$f_{0}\subset f_{1}\subset\cdots\subset f_{n}$$

is called a flag or tower of $P$. We denote the set of all flags of $P$ by $\operatorname{\mathcal{F}l}(P)$. The number of all flags of a polytope $P$ is denoted by $\operatorname{flag}(P)$. Clearly, $f_{n}=P$ and $f_{0}$ is a vertex of $P$.

We establish two recurrence formulae for $\operatorname{flag}(P)$. We define $\phi_{n}:\mathcal{P}^{n}\to\mathbb{R}$ by

$$\phi_{1}(P)=2$$

and

$$\phi_{n}(P)=\sum_{x\in\operatorname{ext}(P)}\phi_{n-1}(P\cap H_{x}),$$

where $H_{x}$ is a hyperplane that separates $x\in{\mathds{R}}^{n}$ strictly from all other extreme points of $P$.

Moreover, we define $\psi_{n}:\mathcal{P}^{n}\to\mathbb{R}$ by

$$\psi_{1}(P)=2$$

and for $n\geq 2$ by

$$\psi_{n}(P)=\sum_{F\in\operatorname{fac}_{n-1}(P)}\psi_{n-1}(F),$$

where $\operatorname{fac}_{n-1}(P)$ is the set of all $n-1$-dimensional faces of $P$.

It can be shown that

$$\phi_{n}(P)=\psi_{n}(P)=\operatorname{flag}(P).$$

We can further prove that for all polytopes with $0$ as an interior point and $P^{\circ}$ its dual/polar polytope

$$\operatorname{flag}(P)=\operatorname{flag}(P^{\circ}),$$

where $P^{\circ}:=\{y\in{\mathds{R}}^{n}\,:\,\forall x\in P:\,\,\langle x,y\rangle\leq 1\}$. It is not difficult to show for the cube $C^{n}$ and the simplex $\Delta^{n}$ in $\mathbb{R}^{n}$

$$\operatorname{flag}(\Delta^{n})=n!\hskip 56.9055pt\operatorname{flag}(C^{n})=\operatorname{flag}((C^{n})^{\circ})=2^{n}n!.$$

For the floating bodies of polytopes we have

Theorem 3.6 was proved by Schütt [58]. Note that Bárány and Larman [9] showed that for all polytopes with an interior point in $\mathbb{R}^{n}$, we have

$$\operatorname{vol}_{n}(P\setminus P_{\delta})\leq c\delta\left(\ln\frac{1}{\delta}\right)^{n-1}.$$

The previous theorem implies fast convergence of the approximation of a given polytope by a polytope with at most $N$ vertices. Indeed, by (3.10), for sufficiently small $\delta\in(0,\infty)$, we have

$$\operatorname{vol}_{n}(P)-\operatorname{vol}_{n}(P_{\delta})\sim\frac{\operatorname{flag}_{n}(P)}{n!n^{n-1}}\delta\left(\ln\frac{1}{\delta}\right)^{n-1}.$$

Therefore, for sufficiently small $\delta\in(0,\infty)$,

$\displaystyle\operatorname{vol}_{n}(P)-\operatorname{vol}_{n}(P_{N})\leq\operatorname{vol}_{n}(P)-\operatorname{vol}_{n}(P_{\delta})\sim\frac{\operatorname{flag}_{n}(P)}{n!n^{n-1}}\delta\left(\ln\frac{1}{\delta}\right)^{n-1}.$

By (3.8)

$$N\leq e^{16n}\frac{\operatorname{vol}_{n}(P)-\operatorname{vol}_{n}(P_{\delta})}{\operatorname{vol}_{n}(B_{2}^{n})\delta}\sim e^{16n}\frac{\operatorname{flag}_{n}(P)}{n!n^{n-1}\operatorname{vol}_{n}(B_{2}^{n})}\left(\ln\frac{1}{\delta}\right)^{n-1}.$$

This implies

$$N^{\frac{1}{n-1}}e^{-32}\left(\frac{n!n^{n-1}\operatorname{vol}_{n}(B_{2}^{n})}{\operatorname{flag}_{n}(P)}\right)^{\frac{1}{n-1}}\leq\ln\frac{1}{\delta}$$

and

$$\delta\leq e^{-N^{\frac{1}{n-1}}}\exp\left(-e^{-32}\left(\frac{n!n^{n-1}\operatorname{vol}_{n}(B_{2}^{n})}{\operatorname{flag}_{n}(P)}\right)^{\frac{1}{n-1}}\right)\sim e^{-N^{\frac{1}{n-1}}}\exp\left(-\frac{e^{-32}n^{\frac{3}{2}}}{(\operatorname{flag}_{n}(P))^{\frac{1}{n-1}}}\right).$$

Since $\delta\left(\ln\frac{1}{\delta}\right)^{n-1}$ is an increasing function for $\delta$ with $0\leq\delta\leq e^{-n+1}$, we obtain

	$\displaystyle\operatorname{vol}_{n}(P)-\operatorname{vol}_{n}(P_{N})$
	$\displaystyle\lesssim\frac{\operatorname{flag}_{n}(P)}{n!n^{n-1}}e^{-N^{\frac{1}{n-1}}}\exp\left(-\frac{e^{-32}n^{\frac{3}{2}}}{(\operatorname{flag}_{n}(P))^{\frac{1}{n-1}}}\right)\left(N^{\frac{1}{n-1}}+\frac{e^{-32}n^{\frac{3}{2}}}{(\operatorname{flag}_{n}(P))^{\frac{1}{n-1}}}\right)^{n-1}.$

This means that the right-hand side expression is of the order

$$\frac{N}{e^{N^{\frac{1}{n-1}}}},$$

which is clearly decreasing rapidly to $0$ for $N\to\infty$.

We now turn to the topic of random approximation. Let $K$ be a convex body in $\mathbb{R}^{n}$. A random polytope in $K$ is the convex hull of finitely many points in $K$ that are chosen at random with respect to a probability measure on $K$. First we consider the natural choice of the normalized Lebesgue measure on $K$, i.e., we consider the uniform distribution on $K$. For a fixed number $N$ of points we are interested in the expected volume of that part of $K$ that is not contained in the convex hull $[x_{1},.....,x_{N}]$ of the chosen points. Let us denote

$$\mathbb{E}(K,N)=\int_{K\times\cdots\times K}\operatorname{vol}_{n}([x_{1},...,x_{N}])\,d\mathbb{P}(x_{1},...x_{N}),$$

(4.11)

where $\mathbb{P}$ is the $N$-fold product of the normalized Lebesgue measure on $K$. We are interested in the asymptotic behavior of

$$\operatorname{vol}_{n}(K)-\mathbb{E}(K,N)=\int_{K\times\cdots\times K}\operatorname{vol}_{n}(K\setminus[x_{1},....,x_{N}])\,d\mathbb{P}(x_{1},...,x_{N}).$$

(4.12)

Rényi and Sulanke [52, 53] proved (4.13) for smooth convex bodies in $\mathbb{R}^{2}$. Wieacker [67] extended their results to higher dimensions for the Euclidean ball. Schneider and Wieacker [56] extended the results to higher dimensions for the mean width instead of the difference volume. It has been solved by Bárány [B] for convex bodies with $C^{3}$ boundary and everywhere positive curvature. The result for arbitrary convex bodies had been shown in [59] (see also [12]).

By choosing the points randomly from the convex body $K$, we get a polytope $P_{N}$ with at most $N$ vertices and symmetric difference distance to $K$ less than

$$\left(\frac{\operatorname{vol}_{n}(K)}{N}\right)^{\frac{2}{n+1}}\frac{1}{2}\left(\frac{n+1}{\operatorname{vol}_{n-1}(B_{2}^{n-1})}\right)^{\frac{2}{n+1}}\frac{(n^{2}+n+2)(n^{2}+1)\Gamma(\frac{n^{2}+1}{n+1})}{(n+3)(n+1)!}\int_{\partial K}\kappa_{K}^{\frac{1}{n+1}}d\mu_{\partial K},$$

while the optimal order is

$$\tfrac{1}{2}\mbox{del}_{n-1}\left(\int_{\partial K}\kappa_{K}(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)\right)^{\frac{n+1}{n-1}}\left(\frac{1}{N}\right)^{\frac{2}{n-1}}.$$

The random approximation is proportinal to $\left(\frac{1}{N}\right)^{\frac{2}{n+1}}$, while the best approximation is proportional to $\left(\frac{1}{N}\right)^{\frac{2}{n-1}}$ by (3.2) and (3.4). On the other hand, as shown in [66], the expected number of vertices of a random polytope equals

$$c_{n}\int_{\partial K}\kappa_{K}(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)\left(\frac{N}{\operatorname{vol}_{n}(K)}\right)^{\frac{n-1}{n+1}},$$

so that the number of vertices of a random polytope is of the order $N^{\frac{n-1}{n+1}}$. This suggests that there exists a random polytope with $N$ vertices whose symmetric difference distance is of the order $(\frac{1}{N})^{\frac{2}{n-1}}$.

It is naturally to expect a better order of random approximation if the points are chosen from the boundary $\partial K$.

Theorem 4.2 has been obtained by Schütt and Werner in [62] proved. Müller [48] proved this result for the Euclidean ball. Reitzner [50] proved this result for convex bodies with $C^{2}$-boundary and everywhere positive curvature.

The condition (4.14) is satisfied if $K$ has a $C^{2}$-boundary with everywhere positive curvature. This follows from Blaschke’s rolling theorem ([10], p.118) and a generalization of it [61]. Indeed, we can choose

$$r=\min_{x\in\partial K}\min_{1\leq i\leq n-1}r_{i}(x)\qquad\text{and}\qquad R=\max_{x\in\partial K}\max_{1\leq i\leq n-1}r_{i}(x),$$

where $r_{i}(x)$ denotes the $i$-th principal curvature radius.

We show now that the expression (4.15) for any other measure given by a density $f$ with respect to $\mu_{\partial K}$ is greater than or equal to the one for (4.17). Since $\int_{\partial K}f(x)d\mu_{\partial K}(x)=1$, we have

	$\displaystyle\left(\frac{1}{\mbox{vol}_{n-1}(\partial K)}\int_{\partial K}\left|\frac{\kappa(x)}{f(x)^{2}}\right|^{\frac{1}{n-1}}d\mu_{\partial K}(x)\right)^{\frac{1}{n+1}}$
	$\displaystyle=\left(\frac{1}{\mbox{vol}_{n-1}(\partial K)}\int_{\partial K}\left|\left(\frac{\kappa(x)}{f(x)^{2}}\right)^{\frac{1}{n^{2}-1}}\right|^{n+1}d\mu_{\partial K}(x)\right)^{\frac{1}{n+1}}\int_{\partial K}f(x)d\mu_{\partial K}(x)$
	$\displaystyle=\left(\frac{1}{\mbox{vol}_{n-1}(\partial K)}\int_{\partial K}\left|\left(\frac{\kappa(x)}{f(x)^{2}}\right)^{\frac{1}{n^{2}-1}}\right|^{n+1}d\mu_{\partial K}(x)\right)^{\frac{1}{n+1}}\times$
	$\displaystyle\hskip 14.22636pt\left(\mbox{vol}_{n-1}(\partial K)\right)^{\frac{2}{n^{2}-1}}\left(\frac{1}{\mbox{vol}_{n-1}(\partial K)}\int_{\partial K}\left|f(x)^{\frac{2}{n^{2}-1}}\right|^{\frac{n^{2}-1}{2}}d\mu_{\partial K}(x)\right)^{\frac{2}{n^{2}-1}}.$

We have $\frac{1}{n+1}+\frac{2}{n^{2}-1}=\frac{1}{n-1}$ and we apply Hölder’s inequality to get

	$\displaystyle\left(\frac{1}{\mbox{vol}_{n-1}(\partial K)}\int_{\partial K}\left|\frac{\kappa(x)}{f(x)^{2}}\right|^{\frac{1}{n-1}}d\mu_{\partial K}(x)\right)^{\frac{1}{n+1}}$
	$\displaystyle\geq\left(\frac{1}{\mbox{vol}_{n-1}(\partial K)}\int_{\partial K}\kappa(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)\right)^{\frac{1}{n-1}}\left(\mbox{vol}_{n-1}(\partial K)\right)^{\frac{2}{n^{2}-1}},$

which gives us

$$\int_{\partial K}\left|\frac{\kappa(x)}{f(x)^{2}}\right|^{\frac{1}{n-1}}d\mu_{\partial K}(x)\geq\left(\int_{\partial K}\kappa(x)^{\frac{1}{n+1}}d\mu_{\partial K}(x)\right)^{\frac{n+1}{n-1}}.$$