Beitr Algebra GeomDOI 10.1007/s13366-013-0153-7

ORIGINAL PAPER

On the families of successive radii and the sumof convex sets

Bernardo González Merino

Received: 5 February 2013 / Accepted: 22 May 2013© The Managing Editors 2013

Abstract In this note we study the behavior of the different families of successiveouter and inner radii with respect to the Minkowski addition of two convex bodies Kand K ′, extending the known cases of the classical successive radii. We get sharp upperand lower bounds for the radii of the sum with respect to the sum of the correspondingradii in most cases.

Keywords Successive radii · Minkowski addition · Projection and section

Mathematics Subject Classification (2000) Primary 52A20; Secondary 52A40

1 Introduction

Let Kn be the set of all convex bodies, i.e., compact convex sets, in the n-dimensionalEuclidean space R

n and let Bn be the n-dimensional Euclidean unit ball.The set of all i-dimensional linear subspaces of R

n is denoted by Łni . For L ∈ Łn

i ,L⊥ denotes its orthogonal complement and for K ∈ Kn and L ∈ Łn

i the orthogonalprojection of K onto L is denoted by K |L . For S ⊂ R

n , we represent by convS itsconvex hull, and we write dim S to denote the dimension of S, i.e., the dimension of itsaffine hull aff S. We use ei for the i-th canonical unit vector in R

n , with lin{u1, . . . , um}we denote the linear hull of the vectors u1, . . . , um and by [u1, u2] we denote the linesegment with end-points u1 and u2.

Supported by MINECO project MTM2012-34037 and by “Programa de Ayudas a Grupos de Excelenciade la Región de Murcia”, Fundación Séneca, 04540/GERM/06.

B. G. Merino (B)Departamento de Matemáticas, Universidad de Murcia, Campus Espinardo, 30100 Murcia, Spaine-mail: bgmerino@gmail.com; bgmerino@um.es

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The diameter, the minimal width, the circumradius and the inradius of a convex bodyK are denoted by D(K ), ω(K ), R(K ) and r(K ), respectively. For more informationon these functionals and their properties we refer to (Bonnesen and Fenchel 1934,pp. 56–59). If f is a functional on Kn depending on the dimension in which a convexbody K is embedded, and if K is contained in an affine subspace A then we writef (K ; A) to denote that f has to be evaluated with respect to the subspace A. With thisnotation we define the following eight families of successive outer and inner radii.

Definition 1.1 Let K ∈ Kn and i ∈ {1, . . . , n}.

Ri (K ) = minL∈Łn

i

R(K |L), ri (K ) = maxL∈Łn

i

maxx∈L⊥

r(K ∩ (x + L); x + L),

Ri (K ) = maxL∈Łn

i

R(K |L), ri (K ) = minL∈Łn

i

maxx∈L⊥

r(K ∩ (x + L); x + L).(1.1)

Observe that Ri (K ) is the smallest radius of a solid cylinder with i-dimensionalspherical cross section containing K , whereas ri (K ) is the radius of the greatesti-dimensional ball contained in K .

If we replace projections by sections in Definition 1.1 (and vice versa) we get fourother series of successive radii.

Definition 1.2 Let K ∈ Kn and i ∈ {1, . . . , n}.˜Ri (K ) = min

L∈Łni

maxx∈L⊥

R(K ∩ (x + L)), ri (K ) = maxL∈Łn

i

r(K |L; L),

Ri (K ) = maxL∈Łn

i

maxx∈L⊥

R(K ∩ (x + L)), ri (K ) = minL∈Łn

i

r(K |L; L).(1.2)

The first systematic study of these successive outer and inner radii can be foundin Betke and Henk (1992). For more information on these radii we refer, both inthe Euclidean or the Minkowski space contexts, for instance, to Betke and Henk(1992, 1993), Boltyanski and Martini (2006), Brandenberg (2005), Brandenberg andTheobald (2006), Eggleston (1958), Gritzmann and Klee (1992), Henk (1992), Henkand Hernández Cifre (2008, 2009) and the references inside.

It is already known (see (Brandenberg and König 2013, Theorem 3.3) that Ri (K ) =Ri (K ) for all i = 1, . . . , n. From the definitions we have that

R(K ) = Rn(K ) = Rn(K ) = ˜Rn(K ),

r(K ) = rn(K ) = rn(K ) = rn(K ) = rn(K ),

D(K )

2= R1(K ) = r1(K ) = r1(K ),

ω(K )

2= R1(K ) = ˜R1(K ) = r1(K ) = r1(K ).

(1.3)

It is clear that all outer successive radii form an increasing sequence in i whereasthe inner successive radii are decreasing in i . Moreover, they are all monotone andhomogeneous functionals of degree 1. We would like to point out that all radii but

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ri , i = 2, . . . , n − 1 (see González and Hernández Cifre 2012, Remark 4.3), arecontinuous functionals in K n with respect to the Hausdorff metric.

The behavior of the diameter, minimal width, circumradius and inradius with respectto the Minkowski sum is well known (see e.g. (Schneider 1993, p. 42)), namely,

D(K + K ′) ≤ D(K ) + D(K ′), ω(K + K ′) ≥ ω(K ) + ω(K ′),R(K + K ′) ≤ R(K ) + R(K ′), r(K + K ′) ≥ r(K ) + r(K ′).

(1.4)

Equality holds, in all cases, for example, when K = K ′. Inequalities (1.4) can betranslated as inequalities for the first and the last of all possible outer and inner radii(see (1.3)). Even more, we have already studied this type of relations in the cases ofthe successive outer and inner radii Ri and ri , namely,

√2 Ri (K + K ′) ≥ Ri (K ) + Ri (K ′), i = 2, . . . , n, (1.5)√2 ri (K + K ′) ≥ ri (K ) + ri (K ′), i = 1, . . . , n − 1., (1.6)

All inequalities are best possible. Moreover, there exist no constants c, c′ > 0 suchthat cRi (K + K ′) ≤ Ri (K ) + Ri (K ′), for i = 1, . . . , n − 1, or c′ri (K + K ′) ≤ri (K )+ ri (K ′), for i = 2, . . . , n (see (González and Hernández Cifre 2012, Theorems1.1 and 1.2)).

In this paper we study the relations between the remaining successive radii describedin (1.1) and (1.2) and the Minkowski sum.

2 Successive radii and Minkowski addition

Theorem 2.1 Let K , K ′ ∈ Kn. Then

˜R1(K + K ′) ≥ ˜R1(K ) + ˜R1(K ′),√2˜Ri (K + K ′) ≥ ˜Ri (K ) + ˜Ri (K ′), i = 2, . . . , n.

All inequalities are best possible.

Proof The lower bound for ˜R1(K + K ′) is well known (see (1.3) and (1.4)). Equalityholds, for instance, if K = K ′.

Let i ∈ {2, . . . , n} and L ∈ Łni be such that

˜Ri (K + K ′) = maxx∈L⊥

R((K + K ′) ∩ (x + L)).

After suitable translations of K and K ′, we may suppose that

R(K ∩ L) = maxx∈L⊥

R(K ∩ (x + L)) and R(K ′ ∩ L) = maxx∈L⊥

R(K ′ ∩ (x + L)).

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The trivial relation (K ∩ L) + (K ′ ∩ L) ⊆ (K + K ′) ∩ L and the monotonicity of thecircumradius imply

˜Ri (K + K ′) = maxx∈L⊥

R((K + K ′) ∩ (x + L)) ≥ R((K + K ′) ∩ L)

≥ R((K ∩ L) + (K ′ ∩ L)).

By (1.5) when i = n, we have

√2R((K ∩ L) + (K ′ ∩ L)) ≥ R(K ∩ L) + R(K ′ ∩ L),

and hence

√2˜Ri (K + K ′) ≥ √

2R((K ∩ L) + (K ′ ∩ L)) ≥ R(K ∩ L) + R(K ′ ∩ L)

= maxx∈L⊥

R(K ∩ (x + L)) + maxx∈L⊥

R(K ′ ∩ (x + L))

≥ ˜Ri (K ) + ˜Ri (K ′).

It remains to show that these inequalities are best possible. We fix i ∈ {2, . . . , n}and consider the convex bodies

K = [−e1, e1] +n

∑

k=i+1

[−ek, ek] and K ′ = [−e2, e2] +n

∑

k=i+1

[−ek, ek].

Here when i = n we are just taking K = [−e1, e1] and K ′ = [−e2, e2]. Since K andK ′ are both (n − i + 1)-dimensional cubes with edges parallel to the coordinate axesand length 2, it is clear that R(K ∩ L), R(K ′ ∩ L) ≥ 1 for all L ∈ Łn

i . Moreover,if L=lin{e1, . . . , ei } then R(K ∩ L) = R(K ′ ∩ L) = 1. This shows that ˜Ri (K ) =˜Ri (K ′) = 1. Now we take the sum

K + K ′ = [−e1, e1] + [−e2, e2] + 2n

∑

k=i+1

[−ek, ek],

an (n − i + 2)-dimensional parallelepiped with edges again parallel to the coordinateaxes and lengths 2 and 4. Then it is easy to see that

˜Ri (K + K ′) = R(

(K + K ′) ∩ lin{e1, . . . , ei }) = √

2 = 1√2

(

˜Ri (K ) + ˜Ri (K ′))

,

which concludes the proof of the theorem. ��

Moreover, ˜Rn(K + K ′) ≤ ˜Rn(K ) + ˜Rn(K ′) (see (1.4)), but there is no reverseinequality for i = 1, . . . , n − 1 (see Theorem 2.6).

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In the case of the outer radius Ri , the result is the following.

Theorem 2.2 Let K , K ′ ∈ Kn. Then

1√2

(

R1(K ) + R1(K ′)) ≤ R1(K + K ′) ≤ R1(K ) + R1(K ′),

√

i(n + 1)

2(i + 1)n

(

Ri (K ) + Ri (K ′)) ≤ Ri (K + K ′) ≤ Ri (K ) + Ri (K ′),

(2.1)

for i = 2, . . . , n. The upper bounds are best possible. The lower bounds are bestpossible in the cases i = 1, n.

Proof We start proving the upper bounds. For i ∈ {1, . . . , n} fixed, let L ∈ Łni be such

that

Ri (K + K ′) = R((K + K ′)|L).

Since (K + K ′)|L = K |L + K ′|L , by (1.4) we have

Ri (K + K ′) = R((K + K ′)|L) = R(K |L + K ′|L) ≤ R(K |L) + R(K ′|L)

≤ Ri (K ) + Ri (K ′).

Equality holds, for instance, if K = K ′.Since R1 = r1, the lower bound for R1(K + K ′) follows from (1.6).Let i ∈ {2, . . . , n} be fixed. The lower bound for Ri (K +K ′) is an easy consequence

of the generalization of Jung’s theorem (see (Henk 1992, Theorem 1)), (1.5) wheni = n and the fact that outer radii Ri form an increasing sequence:

√

n(i + 1)

i(n + 1)Ri (K + K ′) ≥ Rn(K + K ′) = R(K + K ′) ≥ 1√

2(R(K ) + R(K ′))

≥ 1√2

(

Ri (K ) + Ri (K ′))

.

The sharpness in the lower bounds when i = 1, n follows from González andHernández Cifre (2012), Theorem 1.1.

We observe that the left inequality in (2.1) behaves asymptotically as (1.5) when i(and hence n) goes to infinity, since

limi→∞

√

i(n + 1)

2n(i + 1)= 1√

2.

Next we consider the corresponding inequalities for the different families of innerradii.

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Theorem 2.3 Let K , K ′ ∈ Kn. Then

√2r1(K + K ′) ≥ r1(K ) + r1(K ′),2ri (K + K ′) > ri (K ) + ri (K ′) for i = 2, . . . , n − 1 and

rn(K + K ′) ≥ rn(K ) + rn(K ′).

The first and the third inequalities are best possible.

Proof The inequality for the diameter, namely,√

2r1(K + K ′) ≥ r1(K ) + r1(K ′),coincides with (1.6), whereas rn(K + K ′) ≥ rn(K ) + rn(K ′) is well known (see(1.4)).

Let i ∈ {2, . . . , n − 1} be fixed. Since K , K ′ ⊆ K + K ′ (up to translations), then

max{ri (K ), ri (K ′)} ≤ ri (K + K ′)

and hence ri (K ) + ri (K ′) < 2ri (K + K ′).The sharpness in the cases i = 1, n follows from González and Hernández Cifre

(2012), Theorem 1.2. ��Moreover, r1(K +K ′) ≤ r1(K )+r1(K ′) (see (1.4)), but there is no reverse inequality

for i = 2, . . . , n (see Theorem 2.6).

Theorem 2.4 Let K , K ′ ∈ Kn. Then

ri (K + K ′) ≥ ri (K ) + ri (K ′), i = 1, . . . , n.

The inequalities are best possible.

Proof For fixed i ∈ {1, . . . , n}, let L ∈ Łni be such that

ri (K + K ′) = maxx∈L⊥

r((K + K ′) ∩ (x + L); x + L).

After suitable translations of K and K ′ we may suppose without loss of generalitythat

r(K ∩ L; L) = maxy∈L⊥

r(K ∩ (y + L); y + L) and

r(K ′ ∩ L; L) = maxy∈L⊥

r(K ′ ∩ (y + L); y + L).

Now let x ∈ L⊥ be such that

ri (K + K ′) = r((K + K ′) ∩ (x + L); x + L).

The trivial relation

(K ∩ L) + (K ′ ∩ L) ⊆ (K + K ′) ∩ L ,

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together with the monotonicity of r and (1.4), imply that

r((K + K ′) ∩ L; L) ≥ r(K ∩ L; L) + r(K ′ ∩ L; L).

Therefore

ri (K + K ′) = r((K + K ′) ∩ (x + L); x + L) ≥ r((K + K ′) ∩ L; L)

≥ r(K ∩ L; L) + r(K ′ ∩ L; L) ≥ ri (K ) + ri (K ′).

Equality holds, for instance, when K = K ′. ��In the case of Theorem 2.4, as well as for the following results, there exist no reverse

inequalities for all i = 1, . . . , n.

Theorem 2.5 Let K , K ′ ∈ Kn. Then

ri (K + K ′) ≥ ri (K ) + ri (K ′), i = 1, . . . , n.

All inequalities are best possible.

Proof Let i ∈ {1, . . . , n} and L ∈ Łni be such that

ri (K + K ′) = r((K + K ′)|L; L).

By the monotonicity of the inradius in L and (1.4) we get

ri (K + K ′) = r((K + K ′)|L; L) = r(K |L + K ′|L; L)

≥ r(K |L; L) + r(K ′|L; L) ≥ ri (K ) + ri (K ′).

Equality holds, for instance, if K = K ′. ��Next theorem shows the non existence of reverse inequalities for most successive

radii. We write f to denote any of the successive radii ˜Ri for i = 1, . . . , n − 1, ri fori = 2, . . . , n and ri or ri for i = 1, . . . , n.

Theorem 2.6 Let K , K ′ ∈ Kn. Then there exists no constant c > 0 such that

c f (K + K ′) ≤ f (K ) + f (K ′).

Proof Let j ∈ {2, . . . , n} and consider the convex bodies

K j = [−e1, e1] +n

∑

k= j+1

[−ek, ek] and K ′j = [−e2, e2] +

n∑

k= j+1

[−ek, ek],

which have dimension dim K j = dim K ′j = n − j + 1. Here when j = n we are just

taking Kn = [−e1, e1] and K ′n = [−e2, e2].

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First, we deal with ˜Ri . For each i = 1, . . . , n − 1, it is clear that because of thedimension of the sets, Ri (Ki+1) = Ri (K ′

i+1) = 0, and since ˜Ri (K ) ≤ Ri (K ), weget ˜Ri (Ki+1) = ˜Ri

(

K ′i+1

) = 0. However, Ki+1 + K ′i+1 = [−e1, e1] + [−e2, e2] +

2∑n

k=i+2[−ek, ek] contains an (n−i +1)-dimensional cube of edge-length 2 centeredat the origin. Hence, for every L ∈ Łn

i , it holds R((

Ki+1 + K ′i+1

) ∩ L) ≥ 1, and thus˜Ri

(

Ki+1 + K ′i+1

) ≥ 1. It shows the results for ˜Ri .We now consider ri . For any i = 2, . . . , n, it holds ri (Kn−i+2) = ri

(

K ′n−i+2

) = 0,and since Kn−i+2 + K ′

n−i+2 contains an i-dimensional cube of edge-length 2, we haveri

(

Kn−i+1 + K ′n−i+1

) ≥ 1.We finally deal with ri and ri , for i = 1, . . . , n. If i = n, then r(K2) = r

(

K ′2

) = 0and hence rn(K2) = rn(K2) = rn

(

K ′2

) = rn(

K ′2

) = 0. For i ≤ n − 1, let L , L ′ ∈Łn

i be such that L ⊆ e⊥1 and L ′ ⊆ e⊥

2 . Then dim K2|L = dim K ′2|L ′ = i − 1,

which implies that both ri (K2) = ri(

K ′2

) = 0 and therefore ri (K2) = ri(

K ′2

) = 0,and since K2 + K ′

2 contains an n-dimensional cube of edge-length 2, we finally getri

(

K2 + K ′2

) = ri(

K2 + K ′2

) = 1. ��Acknowledgments The author would like to thank the anonymous referee for the very valuable commentsand suggestions, which helped enormously the readability of this paper. I would also like thank MaríaHernández Cifre for fruitful discussions and comments on the subject of this paper.

References

Betke, U., Henk, M.: Estimating sizes of a convex body by successive diameters and widths. Mathematika39(2), 247–257 (1992)

Betke, U., Henk, M.: A generalization of Steinhagen’s theorem. Abh. Math. Sem. Univ. Hamburg 63,165–176 (1993)

Boltyanski, V., Martini, H.: Jung’s theorem for a pair of Minkowski spaces. Adv. Geom. 6, 645–650 (2006)Bonnesen, T., Fenchel, W.: Theorie der konvexen Körper. Springer, Berlin, 1934, 1974. English translation:

of convex bodies. In: Boron, L., Christenson, C., Smith, B. (eds.). BCS Associates, Moscow (1987)Brandenberg, R.: Radii of regular polytopes. Discrete Comput. Geom. 33(1), 43–55 (2005)Brandenberg, R., König, S.: No dimension-independent core-sets for containment under homothetics.

Discrete Comput. Geom. 49(1), 3–21 (2013)Brandenberg, R., Theobald, T.: Radii minimal projections of polytopes and constrained optimization of

symmetric polynomials. Adv. Geom. 6(1), 71–83 (2006)Eggleston, H.G.: Notes on Minkowski geometry. I. Relations between the circumradius, diameter, inradius

and minimal width of a convex set. J. London Math. Soc 33, 76–81 (1958)González, B., Hernández Cifre, M.A.: Successive radii and Minkowski addition. Monatsh. Math. 166(3–4),

395–409 (2012)Gritzmann, P., Klee, V.: Inner and outer j-radii of convex bodies in finite-dimensional normed spaces.

Discrete Comput. Geom 7, 255–280 (1992)Henk, M.: A generalization of Jung’s theorem. Geom. Dedicata 42, 235–240 (1992)Henk, M., Hernández Cifre, M.A.: Intrinsic volumes and successive radii. J. Math. Anal. Appl. 343(2),

733–742 (2008)Henk, M., Hernández Cifre, M.A.: Successive minima and radii. Canad. Math. Bull. 52(3), 380–387 (2009)Schneider, R.: Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge

(1993)

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