The Annals of Statistics2006, Vol. 34, No. 3, 1331–1374DOI: 10.1214/009053606000000335© Institute of Mathematical Statistics, 2006

CONVERGENCE OF ALGORITHMS FOR RECONSTRUCTINGCONVEX BODIES AND DIRECTIONAL MEASURES1

BY RICHARD J. GARDNER, MARKUS KIDERLEN

AND PEYMAN MILANFAR

Western Washington University, University of Aarhusand University of California, Santa Cruz

We investigate algorithms for reconstructing a convex body K in Rn

from noisy measurements of its support function or its brightness functionin k directions u1, . . . , uk . The key idea of these algorithms is to construct aconvex polytope Pk whose support function (or brightness function) best ap-proximates the given measurements in the directions u1, . . . , uk (in the leastsquares sense). The measurement errors are assumed to be stochastically in-dependent and Gaussian.

It is shown that this procedure is (strongly) consistent, meaning that, al-most surely, Pk tends to K in the Hausdorff metric as k → ∞. Here somemild assumptions on the sequence (ui) of directions are needed. Using resultsfrom the theory of empirical processes, estimates of rates of convergence arederived, which are first obtained in the L2 metric and then transferred to theHausdorff metric. Along the way, a new estimate is obtained for the metricentropy of the class of origin-symmetric zonoids contained in the unit ball.

Similar results are obtained for the convergence of an algorithm that re-constructs an approximating measure to the directional measure of a station-ary fiber process from noisy measurements of its rose of intersections in k

directions u1, . . . , uk . Here the Dudley and Prohorov metrics are used. Themethods are linked to those employed for the support and brightness functionalgorithms via the fact that the rose of intersections is the support function ofa projection body.

1. Introduction. The problem of reconstructing an unknown shape from a fi-nite number of noisy measurements of its support function [giving the (signed)distances from some fixed reference point, usually taken to be the origin, to thesupport hyperplanes of the shape] has attracted much attention. The nature ofthe measurements makes it natural to restrict attention to convex bodies. Princeand Willsky [27] used such data in computerized tomography as a prior to im-

Received October 2004; revised July 2005.1Supported in part by NSF Grants DMS-02-03527 and CCR-99-84246, and by the Carlsberg Foun-

dation.AMS 2000 subject classifications. Primary 52A20, 62M30, 65D15; secondary 52A21, 60D05,

60G10.Key words and phrases. Convex body, convex polytope, support function, brightness function,

surface area measure, least squares, set-valued estimator, cosine transform, algorithm, geometrictomography, stereology, fiber process, directional measure, rose of intersections.

1331

1332 R. J. GARDNER, M. KIDERLEN AND P. MILANFAR

prove performance, particularly when only limited data is available. Lele, Kulkarniand Willsky [21] applied support function measurements to target reconstructionfrom range-resolved and Doppler-resolved laser-radar data. The general approachin these papers is to fit a polygon or polyhedron to the data by a least squaresprocedure. In contrast, Fisher, Hall, Turlach and Watson [8] use spline interpola-tion and the so-called von Mises kernel to fit a smooth curve to the data in thetwo-dimensional case. This method was taken up by Hall and Turlach [16] andMammen, Marron, Turlach and Wand [22], the former dealing with convex bod-ies with corners and the latter giving an example to show that the algorithm ofFisher, Hall, Turlach and Watson [8] may fail for a given data set. Further applica-tions and the three-dimensional case can be found in papers by Gregor and Rannou[14], Ikehata and Ohe [18] and Karl, Kulkarni, Verghese and Willsky [19].

Despite all this work, the convergence of even the most straightforward of the re-construction algorithms has apparently never been proved. In Theorem 6.1 below,we provide such a proof for an algorithm we call Algorithm NoisySupportLSQ,due to Prince and Willsky [27]. By convergence, we mean that, given a suitablesequence of directions, the estimators, convex polytopes, obtained by running thealgorithm with noisy measurements taken in the first k directions in the sequence,converge in suitable metrics (the L2 and Hausdorff metrics) to the unknown con-vex body as k tends to infinity. Suitable sequences of directions are those that are“evenly spread,” only slightly more restrictive than the obviously necessary condi-tion that the sequence be dense in the unit sphere.

Moreover, by applying some beautiful and deep results from the theory of em-pirical processes, we are able to provide in Theorem 6.2 estimates of rates ofconvergence of the estimators to the unknown convex body. Some considerabletechnicalities are involved, and some extra conditions are required, of which, how-ever, only a rather stronger condition on the sequence of directions should beregarded as really essential. Convergence rates depend on the dimension of theunknown convex body; for example, for the L2 metric, the rate is of order k−2/5 inthe two-dimensional case, and k−1/3 in the three-dimensional case.

Analogous results are obtained for an algorithm, Algorithm NoisyBrightLSQ,essentially that proposed recently by Gardner and Milanfar [13], that constructsan approximating convex polytope to an unknown origin-symmetric convex bodyfrom a finite number of noisy measurements of its brightness function (giving theareas of the shadows of the body on hyperplanes). The very existence of such analgorithm is highly nontrivial, due to the extremely weak data; each measurementis a single scalar that provides no information at all about the shape of the shadow!Nevertheless, the algorithm has been successfully implemented, even in three di-mensions. Here we are able to prove, for the first time, convergence (Theorem 7.2),with estimates of rates of convergence (Theorem 7.6) also for this algorithm. Onetechnical device that aids in this endeavor is the so-called projection body, whosesupport function equals the brightness function of a given convex body. This al-lows some of our results on reconstruction from support functions to be transferred

CONVERGENCE OF RECONSTRUCTION ALGORITHMS 1333

to the new reconstruction problem. However, we require additional deep results onprojection bodies (a subclass of the class of zonoids) from the theory of convexgeometry due to Bourgain and Lindenstrauss [1] and Campi [4]. Examples of ratesof convergence we obtain, for the Hausdorff metric, are of order k−4/15 in thetwo-dimensional case and k−1/30 in the three-dimensional case.

Most of our results are actually much more informative in that they indicatealso how the convergence depends on the noise level and the scaling of the inputbody. A discussion and the results of some Monte Carlo simulations can be foundin Section 8.

Many auxiliary results are obtained in the course of proving the convergenceof these algorithms, but one is perhaps worth special mention. Roughly speaking,the results we employ from the theory of empirical processes give rates of conver-gence of least squares estimators to an unknown function in terms of the metricentropy of the class of functions involved. In obtaining our results on reconstruc-tion from support functions, it turns out that we therefore need an estimate ofthe metric entropy of the class of compact convex subsets of the unit ball B inn-dimensional space, with the Hausdorff metric. Luckily, the precise order of this,t−(n−1)/2 for sufficiently small t > 0, was previously established by Bronshtein[3] (see Proposition 5.4; it is traditional to talk of ε-entropy rather than t-entropy,but we require ε for a different purpose in this paper). In the problem of recon-struction from brightness functions, however, we need to know the metric entropyof the class of origin-symmetric zonoids contained in B . As far as we know, thisnatural problem has not been addressed before. For n = 2, it is easy to see thatthe answer, t−1/2, is unchanged, but in Theorem 7.3 we show that, for fixed n ≥ 3and any η > 0, the t-entropy of this class is O(t−2(n−1)/(n+2)−η) for sufficientlysmall t > 0. This is somewhat remarkable, since the t-entropy becomes O(t−2)

as n tends to infinity, in complete contrast to the case of general compact convexsets. The hard work behind Theorem 7.3 is done in the highly technical papersof Bourgain and Lindenstrauss [2] and Matoušek [24] on the approximation ofzonoids by zonotopes.

While most of the paper is devoted to reconstruction of convex bodies, Sec-tion 9 focuses on a problem from stereology, that of reconstructing an unknowndirectional measure of a stationary fiber process from a finite number of noisymeasurements of its rose of intersections. It turns out that the corresponding al-gorithm, Algorithm NoisyRoseLSQ, is very closely related to Algorithm Noisy-BrightLSQ, due to the fact that the rose of intersections is the support functionof a projection body. This fact was also used by Kiderlen [20], where an estima-tion method similar to Algorithm NoisyRoseLSQ was suggested and analyzed.Convergence of Algorithm NoisyRoseLSQ was proved by Männle [23], but alsofollows easily from our earlier results (see Proposition 9.1). With suitable extraassumptions, we can once again obtain estimates of rates of convergence of theapproximating measures to the unknown directional measure. These are first givenfor the Dudley metric in Theorem 9.4, but can easily be converted to rates for the

1334 R. J. GARDNER, M. KIDERLEN AND P. MILANFAR

Prohorov metric. For example, for the Prohorov metric, the rate is of order k−1/20

in the three-dimensional case.

2. Definitions, notation and preliminaries. As usual, Sn−1 denotes the unitsphere, B the unit ball, o the origin and ‖ · ‖ the norm in Euclidean n-space Rn. Itis assumed throughout that n ≥ 2. A direction is a unit vector, that is, an elementof Sn−1. If u is a direction, then u⊥ is the (n−1)-dimensional subspace orthogonalto u. If x, y ∈ Rn, then x · y is the inner product of x and y and [x, y] denotes theline segment with endpoints x and y.

If A is a set, dimA is its dimension, that is, the dimension of its affine hull,and ∂A is its boundary. The notation for the usual (orthogonal) projection of A ona subspace S is A|S. A set is origin symmetric if it is centrally symmetric, withcenter at the origin.

We write Vk for k-dimensional Lebesgue measure in Rn, where k = 1, . . . , n,and where we identify Vk with k-dimensional Hausdorff measure. If K is ak-dimensional convex subset of Rn, then V (K) is its volume Vk(K). Define κn =V (B). The notation dz will always mean dVk(z) for the appropriate k = 1, . . . , n.

Let Kn be the family of nonempty compact convex subsets of Rn. A setK ∈ Kn is called a convex body if its interior is nonempty. If K ∈ Kn, then

hK(x) = max{x · y :y ∈ K},for x ∈ Rn, is its support function and

bK(u) = V (K|u⊥),

for u ∈ Sn−1, its brightness function. Any K ∈ Kn is uniquely determined by itssupport function. If K is an origin-symmetric convex body, it is also uniquelydetermined by its brightness function. The Hausdorff distance δ(K,L) betweentwo sets K,L ∈ Kn can be conveniently defined by

δ(K,L) = ‖hK − hL‖∞.

We shall also employ the L2 distance δ2(K,L) defined by

δ2(K,L) = ‖hK − hL‖2.

By Proposition 2.3.1 of [15], there is a constant c = c(n) such that if K and L arecontained in RB for some R > 0, then

δ(K,L) ≤ cR(n−1)/(n+1)δ2(K,L)2/(n+1),(1)

which shows (together with a trivial inequality in the converse direction) that bothmetrics induce the same topology on Kn.

A zonotope is a vector sum of finitely many line segments. A zonoid is the limitin the Hausdorff metric of zonotopes. The projection body of a convex body K

in Rn is the origin-symmetric convex body �K defined by

h�K = bK.

CONVERGENCE OF RECONSTRUCTION ALGORITHMS 1335

An introduction to the theory of projection bodies is provided by Gardner [10],Chapter 4. It turns out that projection bodies are precisely the n-dimensionalorigin-symmetric zonoids. For this reason, we shall denote the set of projectionbodies in Rn by Zn.

The surface area measure S(K, ·) of a convex body K is defined for Borelsubsets E of Sn−1 by

S(K,E) = Vn−1(g−1(K,E)

),(2)

where g−1(K,E) is the set of points in ∂K at which there is an outer unit nor-mal vector in E. The convex body P is a zonotope if and only if P = �K forsome origin-symmetric convex polytope K . In this case, S(K, ·) is a sum of pointmasses, each located at one of the directions of the line segments whose sum is P

and with weight equal to half the length of this line segment. This fact will be usedin a reconstruction algorithm in Section 7.

A fundamental result is Minkowski’s existence theorem (see, e.g., [10], Theo-rem A.3.2), which says that a finite Borel measure µ in Sn−1 is the surface areameasure of some convex body K in Rn, unique up to translation, if and only if µ

is not concentrated on any great sphere and∫Sn−1

udµ(u) = o.

The treatise of Schneider [28] is an excellent general reference for all of thesetopics.

Let U = {u1, . . . , uk} ⊂ Sn−1. The nodes corresponding to U are defined asfollows. The hyperplanes u⊥

i , i = 1, . . . , k, partition Rn into a finite set of poly-hedral cones, which intersect Sn−1 in a finite set of spherically convex regions.The nodes ±vj ∈ Sn−1, j = 1, . . . , l, are the vertices of these regions. Thus, whenn = 2, the nodes are simply the 2k unit vectors each of which is orthogonal tosome ui , i = 1, . . . , k. When n = 3, each vj is of the form (ui × ui′)/‖ui × ui′‖,where 1 ≤ i < i′ ≤ k and ui = ±ui′ . Thus, for n = 3, l ≤ k(k−1)/2 and in general,l = O(kn−1). Campi, Colesanti and Gronchi [5] proved the following result.

PROPOSITION 2.1. Let K be a convex body in Rn and let U = {u1, . . . , uk} ⊂Sn−1 span Rn. Among all convex bodies with the same brightness function valuesas K in the directions in U , there is a unique origin-symmetric convex polytope P ,of maximal volume and with each of its facets orthogonal to one of the nodescorresponding to U .

This implies that, for any projection body �K and any finite set of directionsU ⊂ Sn−1, there is a zonotope Z with hZ(u) = h�K(u), for all u ∈ U . Moreover,Z can be written as a sum of line segments, each parallel to some node correspond-ing to U .

1336 R. J. GARDNER, M. KIDERLEN AND P. MILANFAR

The following deep result was proved independently by Campi [4] (for n = 3)and Bourgain and Lindenstrauss [1]. The latter authors state their theorem in termsof a metric other than the Hausdorff metric, and make an additional assumptionon the distance between the projection bodies. Groemer ([15], Theorem 5.5.7)presents the version below, and his proof yields the estimate of the constant in (4).This involves some tedious calculations (see www.ac.wwu.edu/~gardner; no at-tempt was made to obtain the optimal expression). In (4) and throughout the pa-per, the “big O” notation is used in the sense of “less than a constant multipledepending only on n.”

PROPOSITION 2.2. Let K and L be origin-symmetric convex bodies in Rn,n ≥ 3, such that

r0B ⊂ K,L ⊂ R0B,

for some 0 < r0 < R0. If 0 < a < 2/(n(n + 4)), there is a constant c =c(a,n, r0,R0) such that

δ(K,L) ≤ cδ2(�K,�L)a.(3)

Moreover, if 0 < a < 2/(n(n + 4)) is fixed, r0 < 1 and R0 > 1, then

c = O(r−2n−10 R5

0).(4)

3. Some properties of sets and sequences of unit vectors. In this section wegather together some basic results on sets and sequences of unit vectors that willbe useful in Sections 5 and 7.

If {u1, . . . , uk} is a finite subset of Sn−1, its spread �k is defined by

�k = maxu∈Sn−1

min1≤i≤k

‖u − ui‖.(5)

For i = 1, . . . , k, let �i be the spherical Voronoi cell

�i = {u ∈ Sn−1 :‖u − ui‖ ≤ ‖u − uj‖ for all 1 ≤ i, j ≤ k}(6)

containing ui . Then⋃k

i=1 �i = Sn−1, and we define

ωk = max1≤i≤k

Vn−1(�i).(7)

By the definition of spread, {u1, . . . , uk} is a �k-net in Sn−1, meaning that, forevery vector u in Sn−1, there is an i ∈ {1, . . . , k} such that u is within a distance �k

of ui . The existence of ε-nets in Sn−1 with relatively few points is provided by thefollowing well-known result. It can be proved by induction on n in a constructiveway; see, for example, [13], Lemma 7.1.

PROPOSITION 3.1. For each ε > 0 and n ≥ 2, there is an ε-net in Sn−1 con-taining O(ε1−n) points.

CONVERGENCE OF RECONSTRUCTION ALGORITHMS 1337

Now let (ui) be an infinite sequence in Sn−1. We retain the notation �k forthe spread of the first k points in the sequence, and similarly for ωk . We need toconsider some conditions on (ui) that are stronger than denseness in Sn−1. To thisend, for u ∈ Sn−1 and 0 < t ≤ 2, let

Ct(u) = {v ∈ Sn−1 :‖u − v‖ < t}be the open spherical cap with center u and radius t . We call (ui) evenly spread iffor all 0 < t < 2, there is a constant c = c(t) > 0 and an N = N(t) such that

|{u1, . . . , uk} ∩ Ct(u)| ≥ ck,(8)

for all u ∈ Sn−1 and k ≥ N .The following lemma provides relations between various properties of se-

quences we need later. A discussion of how these properties relate to the well-known concept of a uniformly distributed sequence can be found in the Appendixof [11].

LEMMA 3.2. Consider the following properties of a sequence (ui) in Sn−1:

(i) �k = O(k−1/(n−1)).(ii) ωk = O(k−1) and (ui) is dense in Sn−1.

(iii) (ui) is evenly spread.(iv) (ui) is dense in Sn−1.

Then (i) ⇒ (ii) ⇒ (iii) ⇒ (iv), and there are sequences with property (i).

PROOF. Assume (i), and let k ∈ N and i ∈ {1, . . . , k}. Let �i , 1 ≤ i ≤ k, bethe Voronoi cells corresponding to the set {u1, . . . , uk} defined by (6). Note that�i ⊂ C�k

(ui) and hence,

Vn−1(�i) ≤ Vn−1(C�k

(ui)) ≤ Vn−1(Dk(ui)),

where Dk(ui) is the (n − 1)-dimensional ball in the tangent hyperplane to Sn−1

at ui , obtained by the inverse spherical projection (with center o) of C�k(ui). If

�k <√

2, then Dk(ui) has center ui and radius rk = tan(2 arcsin(�k/2)). There-fore,

ωk = max1≤i≤k

Vn−1(�i) ≤ rn−1k κn−1 = O(�n−1

k ) = O(k−1).

Since it is clear that (i) also implies that (ui) is dense in Sn−1, (ii) holds.Suppose that (ii) holds. Fix 0 < t < 2 and u ∈ Sn−1. Cover Sn−1 with finitely

many open caps Cj = Ct/6(vj ), 1 ≤ j ≤ m. Since (ui) is dense in Sn−1, there is anN = N(t) ∈ N such that, for k ≥ N , any of these caps contains at least one point of{u1, . . . , uk}. The cap Ct/3(u) contains at least one Cj , and hence a point ui0 with1 ≤ i0 ≤ N . Note that N does not depend on u.

1338 R. J. GARDNER, M. KIDERLEN AND P. MILANFAR

Fix k ≥ N and let �i , 1 ≤ i ≤ k, be the Voronoi cells corresponding to the set{u1, . . . , uk}. If �i ∩ intCt/3(u) = ∅, i = i0, there must be a point in Ct/3(u) closerto ui than to ui0 . This implies ui ∈ Ct(u). Consequently,

intCt/3(u) ⊂ ⋃{�i :�i ∩ intCt/3(u) = ∅} ⊂ ⋃{�i :ui ∈ Ct(u)}.Now (ii) implies that there is a c′ = c′(t) such that

Vn−1(Ct/3(u)) ≤ ∑ui∈Ct (u)

Vn−1(�i)

≤ ωk|{i :ui ∈ Ct(u)}|≤ c′

k|{u1, . . . , uk} ∩ Ct(u)|.

Since the left-hand side of the previous chain of inequalities does not depend on u,this yields (iii). That (iii) implies (iv) is clear.

To obtain a sequence with property (i), observe that, by Proposition 3.1, there isa constant C such that, for each m ∈ N, there is a set Wm of at most C2m(n−1) unitvectors forming a 2−m-net. Order the elements of each Wm in an arbitrary fashion,and let (ui) be the sequence obtained by forming one sequence from these finitesequences W1, W2 and so on in that order. Let

Nm = C(2n−1 + 22(n−1) + · · · + 2m(n−1)) = C2n−1

(2m(n−1) − 1

2n−1 − 1

).

Then for all k ≥ Nm, the points u1, . . . , uk form a 2−m-net.Now suppose that k is the least integer such that the points u1, . . . , uk have

spread �k , where

2−m ≤ �k < 21−m.

Then

k ≤ Nm = C2n−1(

2m(n−1) − 1

2n−1 − 1

)< C2n−1

(2n−1�1−n

k − 1

2n−1 − 1

),

or

�k ≤(

k(2n−1 − 1)

C22(n−1)+ 1

2n−1

)−1/(n−1)

= O(k−1/(n−1)). �

Let (ui) be a sequence of vectors in Sn−1. For application in Section 7, we needto consider properties of the “symmetrized” sequence

(u∗i ) = (u1,−u1, u2,−u2, . . .).(9)

Let

�∗k = max

u∈Sn−1min

1≤i≤kmin{‖u − ui‖,‖u − (−ui)‖}(10)

CONVERGENCE OF RECONSTRUCTION ALGORITHMS 1339

be the symmetrized spread of u1, . . . , uk . Also, let ω∗k be the maximum Vn−1-mea-

sure of the 2k spherical Voronoi cells corresponding to the set {±u1,±u2, . . . ,

±uk}.Following [20], page 14, we call (ui) asymptotically dense if

lim infk→∞

1

k|{u1, . . . , uk} ∩ G| > 0,

for all origin-symmetric open sets G = ∅ in Sn−1.

LEMMA 3.3. Consider the following properties of a sequence (ui) in Sn−1:

(i) �∗k = O(k−1/(n−1)).

(ii) ω∗k = O(k−1) and (u∗

i ) is dense in Sn−1.(iiia) (u∗

i ) is evenly spread.(iiib) (ui) is asymptotically dense.

(iv) (u∗i ) is dense in Sn−1.

Then (i) ⇒ (ii) ⇒ (iiia) ⇔ (iiib) ⇒ (iv), and there are sequences with prop-erty (i).

PROOF. The implications (i) ⇒ (ii) ⇒ (iiia) ⇒ (iv) are direct consequencesof Lemma 3.2 and the definition of (u∗

i ). The existence statement also fol-lows from this lemma, as any sequence with �k = O(k−1/(n−1)) satisfies �∗

k =O(k−1/(n−1)).

That (iiia) implies (iiib) is trivial. To prove the converse, let Ct(u) be an opencap in Sn−1 of radius t , and cover the compact set Sn−1 with open caps C1, . . . ,Cm

of radius t/2. Then Cj ⊂ Ct(u) for some j . If (ui) is asymptotically dense, we canapply the definition of this property with G = Cj ∪ (−Cj) to conclude that thereare a constant c′ > 0 and an N ′ such that∣∣{u1, . . . , uk} ∩ (

Cj ∪ (−Cj))∣∣ ≥ c′k

for all k ≥ N ′ and, hence, that

|{±u1, . . . ,±uk} ∩ Ct(u)| ≥ c′k

for all k ≥ N ′. From this, it follows easily that (u∗i ) is evenly spread. �

4. Metric entropy and convergence rates for least squares estimators. LetG = ∅ be a class of measurable real-valued functions defined on a subset E of Rn.Suppose that xi ∈ E, i = 1,2, . . . , are fixed, and let Xi , i = 1,2, . . . , be indepen-dent random variables with mean zero and finite variance. If g0 ∈ G, we regard thequantities

yi = g0(xi) + Xi,

1340 R. J. GARDNER, M. KIDERLEN AND P. MILANFAR

i = 1,2 . . . , as measurements of the unknown function g0. For k ∈ N, any functiongk ∈ G satisfying

gk = arg ming∈G

k∑i=1

(yi − g(xi)

)2(11)

is called a least squares estimator for g0 with respect to G, based on measurementsat x1, . . . , xk . (Since gk depends on y1, . . . , yk , it also depends on the random vari-ables X1, . . . ,Xk , but this is not made explicit.) If k, G and x1, . . . , xk are clearfrom the context, we shall simply refer to gk as a least squares estimator for g0.In the definition of gk , xi and yi are not needed for i > k, but later we shall takeadditional measurements into account in order to examine the asymptotic behav-ior of gk as k increases. In general, gk need not be unique and the existence of aleast squares estimator has to be assumed. In the applications we have in mind,a least squares estimator always exists. To provide the necessary measurability forthe background theory to work, a suitable condition can be imposed on the class G.Following [25], page 196, we call G permissible if it is indexed by a set Y that isan analytic subset of a compact metric space, such that G = {g(·, y), y ∈ Y }, andg(·, ·) : Rn ×Y → R is Ln ⊗B(Y )-measurable, where Ln is the class of Lebesguemeasurable sets in Rn and B(Y ) is the class of Borel subsets of Y . The metric on Y

will be important only insofar as it determines B(Y ).Let (S, d) be a set S equipped with a pseudometric d and let ε > 0. A set U ⊂ S

is called an ε-net if each point in S is within a d-distance at most ε of some pointin U .

We can now define metric entropy, a valuable concept introduced byKolmogorov. Metric entropy is often also called ε-entropy, but we need to re-serve the letter ε for a different purpose. Accordingly, we define the t-coveringnumber N(t, S, d) of (S, d) as the least cardinality of all t-nets. In other words,N(t, S, d) is the least number of balls of radius t with respect to d that cover S.Then H(t, S, d) = logN(t, S, d) is called the t-entropy of (S, d), and we can dropthe argument d when there is no possibility of confusion. This notion will mainlybe used for subsets of G. For k ∈ N, we define a pseudonorm | · |k on G by

|g|k =(

1

k

k∑i=1

g(xi)2

)1/2

, g ∈ G.

Note that this pseudonorm depends on x1, . . . , xk . For ε > 0, let

Gk(ε, g0) = {g ∈ G : |g − g0|k ≤ ε}.Then we denote by H(t,Gk(ε, g0)) the t-entropy of Gk(ε, g0) with respect to thepseudometric generated by the pseudonorm | · |k ; again, this depends on x1, . . . , xk .If G is a cone (i.e., G = sG for all s > 0), then

H(t,Gk(ε, g0)

) = H(st,Gk(sε, sg0)

) = H(st, sGk(ε, g0)

),(12)