Floating bodies and approximation of convex bodies by polytopes ∗

: We describe some results on approximation of convex bodies by polytopes. Best and random approximations are considered and compared. The geometric quantities related to the convex body, that appear naturally in such approximation questions are the aﬃne surface areas. Those, and their relation to ﬂoating bodies, will be discussed as well. In this survey we give only a very selective collection of results on ap- proximation by polytopes.


Introduction
How well can a convex body be approximated by a polytope? This is a central question in the theory of convex bodies, not only because it is a natural question and interesting in itself but also because it is relevant in many applications, for instance in computer vision, tomography, geometric algorithms, local theory of Banach spaces, stochastic geometry and many more. We only quote [4,9,10,11,13,15,17,18,19,26,30].
It often involves side conditions like a prescribed number of vertices, or, more generally, k-dimensional faces and a requirement that the body contains the polytope or vice versa. Various metrics are used to measure accuracy of approximation. Ideally, one then wants the optimal dependence on all the parameters involved: the dimension n, the convex body K, the number of prescribed vertices or facets or k-faces. An example of such a result of asymptotic nature is the following remarkable theorem due to Gruber [13] in dimension n and due to McClure and Vitale [24] in dimension 2 which says that for a convex body K in R n with sufficiently smooth boundary inf{d s (K, P N )| P N is a polytope, contained in K, having at most N vertices} is asymptotically equal as N → ∞ to (1) Here d s is the symmetric difference metric (see below), ∂K denotes the boundary of the convex body K, μ K is the usual surface area measure on ∂K and κ is the Gauss curvature. This result tells us that when we approximate by an inscribed polytope with a fixed number of vertices N the optimal dependence on N in this case is N −2/(n −1) . The optimal dependence on the dimension is hidden in the constant del n−1 . Only later it was determined that del n−1 is of the order √ n, [12,22,23]. The dependence on the body K comes in via the affine surface area ∂K κ(x) 1 n+1 dμ K (x). This quantity, in a sense, "measures" the boundary behavior of a convex body, so it is natural that it should appear in questions of approximation of convex bodies by polytopes. In Section 2 we describe that quantity and its generalizations in more detail.
It is only in rare special cases that a best approximating polytope can be explicitly singled out. Consequently, a common practice is to randomize: choose N points at random in the convex body with respect to a probability measure P. The convex hull of these randomly chosen points is called a random polytope.
As is demonstrated in Section 4, choosing the points randomly inside the convex body does not yield optimal dependence with respect to the number of chosen points. It is more economical to choose the points directly from the boundary of the body.
In Section 4 we quote a result by Schütt and Werner [35], which, in a nutshell, can be phrased as "random approximation is as good as best approximation". More precisely, let E(K, N ) be the expected volume of a random polytope, the convex hull of N randomly chosen points on ∂K. It is proved in [35] that there is a constant c n such that (2) We will comment in Section 4 on the constant c n and its relation to del n−1 .
Here, and elsewhere, vol n (K) denotes the volume of K. We observe that again affine surface area appears. Note also that affine surface area is a priori only defined for sufficiently smooth bodies so that the Gauss curvature exits. This is a drawback as we want such approximation results for all convex bodies. It is exactly here that floating bodies, and their variants, surface bodies and weighted floating bodies come in. These bodies can be used to define affine surface area and L p affine surface area for all convex bodies. This will be explained in Section 3.
In this survey we give only a very selective collection of results on approximation by polytopes. For complementary reading we suggest the paper [27].

Definitions, examples and properties
Throughout this note we will assume without loss of generality that the center of gravity or centroid of a convex body K in R n is at the origin. For real p = −n, the L p affine surface area as p (K) of K was introduced for p > 1 in a seminal paper [21] by Lutwak and extended to all p = −n in [36], and where ·, · is the standard inner product on R n which induces the Euclidian norm · and N (x) is the outer unit normal vector at x ∈ ∂K. In particular, for p = 0 we get The case p = 1 is the classical affine surface area which goes back to Blaschke [5]. A definition for p = −n was proposed in [25]. If the boundary of K is sufficiently smooth then (3) and (4) can be written as integrals over the boundary ∂B n 2 = S n−1 of the Euclidean unit ball B n 2 in R n , Here, σ is the usual surface area measure on S n−1 , h K (u) = max x∈K x, u is the support function of direction u ∈ S n−1 , and f K (u) is the curvature function, i.e. the reciprocal of the Gaussian curvature κ(x) at this point x ∈ ∂K that has u as outer normal. In particular, for p = ±∞, where K • = {y ∈ R n , x, y ≤ 1, ∀x ∈ K} is the polar body of K. As the Gauss curvature of a polytope is 0 almost everywhere, the L p affine surface area of a polytope is 0 or ∞, depending on p. Other examples are listed next. The details can be found in [36]. .
In particular, for the Euclidean unit ball B n 2 , for all p = −n, The next theorem collects some of the properties of L p affine surface area. These properties make affine surface area a useful instrument in questions where information on the boundary of a convex body is needed. In particular, the affine isoperimetric inequalities, which are stronger than their Euclidean counterparts, are an efficient tool to detect ellipsoids.

Theorem 2. Let K be a convex body in R n . (i) For all p = −n and for all invertible linear transformations
(ii) For 0 ≤ p ≤ ∞, L p affine surface area is an upper semicontinuous functional and for −n < p ≤ 0 a lower semicontiuous functional with respect to the Hausdorff metric.

Equality holds in both inequalities iff K is an ellipsoid. Equality holds trivially in both inequalities if
Property (i) was shown in [21,36]. The semicontinuity is due to Lutwak [21]. The L p affine isoperimetric inequalities were proved by Lutwak [21] for p > 1 and for all other p by Werner and Ye [39]. The case p = 1 is the classical case. For −∞ ≤ p < −n there is an inequality as well and it was proved in [39].

Rényi divergenes of cone measures
Let (X, μ) be a measure space and let dP = pdμ and dQ = qdμ be probability measures on X that are absolutely continuous with respect to the measure μ. Then the Rényi divergence of order α, introduced by Rényi [29], is defined for α = 1 as The integrals are also called Hellinger integrals, see e.g. [16] for those integrals and additional information.
Usually in the literature, the measures are probability measures. Therefore we normalize the measures. An important special case of Rényi divergence is the case when α → 1, which leads to the Kullback-Leibler divergence or relative entropy from P to Q (see [8]), In [38] Rényi divergence was introduced for convex bodies K in R n , as follows. Let 118

E. M. Werner
Then are probability measures on ∂K that are absolutely continuous with respect to μ K . These measures can be viewed as the cone measures of the convex bodies K and K • , respectively. We refer to [38] for the details (see also Section 4.2.2). We then can define the Rényi divergence of K of order α for all α.

Definition 3. [38]
Let K be a convex body in R n and let α ∈ R, α = 1. Then the Rényi divergences of order α of K are The L p affine surface areas are a central part of the L p -Brunn Minkowski theory, an extension of the classical Brunn Minkowski theory, see, e.g., [31]. A remarkable fact which connects L p -Brunn Minkowski theory and information theory was observed in [38], namely: L p affine surface areas of a convex body are exponentials of Rényi divergences of the cone measures of K and K • .
In particular,

Definitions and properties
For a convex body K ⊆ R n and 0 ≤ δ < voln(K) 2 , the floating body K δ was introduced independently by Barany and Larman [3] and by Schütt and Werner [34], as the intersection of all halfspaces H + whose defining hyperplanes H cut off a set of volume δ from K, It is obvious that K δ ⊆ K, that K δ is convex and that K 0 = K.
Extensions of the concept of floating body to spherical and hyperbolic space can be found in [6,7].
Both, the surface body and the weighted floating body, which we introduce next, are variants of the floating body. The surface body and the weighted floating body are relevant in approximation of convex bodies by polytopes.
The surface body was introduced in [36].
The surface body K f,s is the intersection of all the closed half-spaces H + whose defining hyperplanes H cut off a set of P f -measure less than or equal to s from ∂K. More precisely, It follows from the Hahn-Banach theorem that K f,0 ⊆ K. If in addition f is almost everywhere nonzero, then K f,0 = K.
Another variant of the floating body is the weighted floating body, introduced in [37]. In the following definition m is the Lebesgue measure on R n . Definition 6. [37] Let s > 0 and let f : K → R be a nonnegative, integrable function with K fdm = 1, where m is the Lebesgue measure on R n .
The weighted floating body F (K, f, s) is the intersection of all the closed halfspaces H + whose defining hyperplanes H cut off a set of (f · m)-measure less than or equal to s from K. More precisely,

Surface body and L p affine surface area
As noted, a priori the L p affine surface areas are only defined for sufficiently smooth bodies. It was shown in [34] that for any convex body K in R n , where κ is the generalized Gauss curvature, see e.g., [35]. Thus the right hand side of this equation can be used as a definition of affine surface area which is now valid for all convex bodies.
Using the surface body or the weighted floating body, extensions of L p affine surface area to general convex bodies can be achieved. The crucial step to do this is the next theorem, for which we need the notion of rolling function r which was introduced in [34] as if K has a unique normal at x. If K does not have a unique normal at x then r(x) = 0. The rolling function allows to give a quantitive version of Blaschke's rolling theorem which states that for every convex body K with C 2 boundary and everywhere strictly positive and bounded Gaussian curvature, there is a Euclidean ball with sufficiently small radius r that can roll freely inside the convex body K. The latter means that for any point x ∈ ∂K there is y ∈ K such that x ∈ B n 2 (y, r) and B n 2 (y, r) ⊆ K.
The quantitative version of this theorem is next.

Theorem 7. [34] Let K be a convex body in R n such that it contains B n 2 . Then we have for all t with 0 ≤ t ≤ 1, that {x ∈ ∂K|r(x) ≥ t} is a closed set and
The inequality is optimal.
Another notion that is needed is that of minimal function, which was introduced in [36]. Let f : ∂K → R be an integrable, almost everywhere strictly positive function. For x ∈ ∂K and s > 0 we put is the minimal function. Here, H(x s , N Ks (x s )) is the hyperplane through x s with normal N Ks (x s )).

Theorem 8. [36]
Let K be a convex body in R n . Suppose that f : ∂K → R is an integrable, almost everywhere strictly positive function such that fdμ = 1. The next corollary shows that the surface body can be used to extend the definition of L p affine surface area to all convex bodies. To do so, we define for q, −∞ ≤ q ≤ ∞, q = −n the functions f q : ∂K → R as follows. For q = ±∞, put (17) and for all other values of q

Assume that
Corollary 9. [36] Let K be a convex body in R n with the origin in its interior.
and for p = −1 let q = ∞. Let f q be as in (17) and (18) and assume that it is almost everywhere strictly positive. Assume that Weighted floating bodies can similarly be used to provide geometric interpretations of L p affine surface area. We only quote

The main theorem
Ideally for applications one seeks an algorithm that produces, for a given convex body, a best approximating (in a given metric) polytope. The works by e.g., Gordon, Meyer and Reisner [11], by Lopez and Reisner [17] and by Schütt [33] provide constructions of such algorithms. We want to note that the algorithm in [33] is based on the floating bodies of Section 3. The typical questions in this context are: (i) What is the order of magnitude of the best approximation of a convex body K in R n by a polytope P with a fixed number of vertices?
We will concentrate on this aspect. Moreover, here we will only consider approximating polytopes that are inscribed in the body K. Arbitrary positioned approximating polytopes were considered in [14,20]. If one considers such a setup, then one gains by a factor of dimension. (ii) What is the order of magnitude of the best approximation of a convex body K in R n by a polytope P with a fixed number of (n−1)-dimensional faces or -more generally -by a fixed number of k-dimensional faces, 1 ≤ k ≤ n − 1?
For results on that item we refer to the literature. Of course we have to specify which metric we use in these approximation questions. Here, we only consider the symmetric difference metric d s , which, for two convex bodies K and L in R n is given by There are many other metrics that have been considered. Again we refer to the literature for more on that topic. Many of the results deal with asymptotic estimates. A typical example is given by the result (1) by Gruber [13] and McClure and Vitale [24] mentioned in the introduction.
But it is only in rare special cases that a best approximating polytope can be explicitly singled out. Consequently, a common practice is to randomize. Consider the random polytope P N obtained as the convex hull of N points chosen at random in the convex body with respect to a probability measure P. A natural choice for the probability measure P is the normalized Lebesgue measure on K. Denote the expected volume of a random polytope of N randomly chosen points by E (K, N ).
A striking example of an asymptotic result of a probabilistic nature is due to Bárány [2] (in case the boundary is C 3 ), and Schütt [32] in the general case: where c > 0 is an absolute constant. Note that the order of magnitude N −2/(n+1) of the random result is not as good the best order of approximation provided by (1). But this is to be expected: not all points chosen inside K are necessarily vertices of the approximating random polytope.
A more economical way is to choose the points at random with respect to a probability measure directly on the boundary of the body.
The setting is now as follows. For an integrable, nonnegative function f : ∂K → R with ∂K f (x)dμ K = 1 we denote by P f the probability measure with dP f = fdμ K . We considerer random polytopes P N where the points are chosen from the boundary of K with respect to P f . Then the expected volume of such a random polytope is where [x 1 , . . . , x N ] is the convex hull of the points x 1 , . . . , x N . For such a setting the following theorem was proved in [35]. A crucial ingredient in the proof of this theorem were the surface bodies. This theorem was also proved in [28] under stronger smoothness assumptions on the boundary of K. Theorem 11. [35] Let K be a convex body in R n such that there are r and R in R with 0 < r ≤ R < ∞ so that we have for all x ∈ ∂K and let f : ∂K → R + be a continuous, positive function with ∂K f (x)dμ ∂K (x) = 1. Let P f be the probability measure on ∂K given by where κ is the (generalized) Gauß-Kronecker curvature and The minimum at the right hand side is attained for the normalized affine surface area measure with density .
As the random polytope P N ⊆ K, all the above results are approximation results in the symmetric difference metric. The condition: there are r and R in R with 0 < r ≤ R < ∞ so that we have for all x ∈ ∂K is satisfied if K has a C 2 -boundary with everywhere positive curvature. This follows from Blaschke's rolling theorem [5], respectively it's generalization, Theorem 7, of [34]. Indeed, we can choose where r i (x) denotes the i-th principal curvature radius. By a result of Alexandroff [1], the generalized curvature κ exists a.e. on a convex body. It was shown in [34] that κ 1 n+1 is an integrable function. Therefore the density exists provided that ∂K κ(x) 1 n+1 dμ K (x) > 0. This is certainly assured by the assumption (20) on the boundary of K.
Hence, by this theorem, we get the best random approximation if we choose the points on the boundary of K with respect to the affine surface area measure P fas . Then the order of magnitude for this random approximation is It is natural to see how this best random approximation compares with best approximation.

Comparison: Best approximation and random approximation
Recall that by (1) best approximation is of the order Thus, to see how random approximation (22) compares to the best approximation (23), we only need to compare c n and 1 2 del n−1 . c n is given by (21). del n−1 was determined in a series of papers by Gordon, Reisner and Schütt [12] and Mankiewicz and Schütt [22,23]. Using these results, one gets that with an absolute constant c > 0, Therefore, surprisingly, random approximation is (almost) as good as best approximation.

Other measures P f
Aside from the best approximating measure P fas , there are other measures of interest. We list some of them. 1. The second measure of interest is the surface measure given by the constant density This measure is not affine invariant and we get Floating bodies, random polytopes 125 2. The next measure is the normalized cone measure, see also (10). Recall that we assume throughout without loss of generality that 0 is the center of gravity of a convex body K. Let A be a subset of ∂K and denote by [0, A] the convex hull of 0 and A. Then the normalized cone measure is given by As 1 n ∂K x, N (x) dμ K (x) = vol n (K), its density is where q k is as in (10). The measure is invariant under linear, volume preserving maps. We get Recalling the L p affine surface area of Section 2, we observe that the expression on the right hand side of (24) is exactly as p=n/(n−2) . Thus one is naturally led to suppose that other L p affine surface areas might appear in the setting of Theorem 11.

Random polytopes and L p affine surface areas
We want to present a geometric characterization of the L p affine surface area for all p similar in spirit to the one given in Corollary 9.
For q, −∞ ≤ q ≤ ∞, q = −n, let the functions f q : ∂K → R be given as in (17) and (18). Then the following corollary is an immediate consequence of Theorem 11. Corollary 12. Let K be a convex body in R n with the origin in its interior. Assume also that there are r and R in R with 0 < r ≤ R < ∞ so that we have for all x ∈ ∂K

E. M. Werner
Thus Corollary 12 can be viewed as a geometric interpretation, in the spirit of Corollary 9, of L p affine surface areas via random polytopes.