ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE Abstract. We study the behavior of the so called successive inner and outer radii with respect to the p-sums of convex bodies, p 1, which were introduced by Firey in the early 60 s. These results generalize the corresponding relations for the classical Minkowski addition. We get all possible upper and lower bounds for the radii of the p-sum of two convex bodies in terms of the corresponding radii. 1. Introduction Let K n be the set of all convex bodies, i.e., compact convex sets, in the n-dimensional Euclidean space R n. Let, and be the standard inner product and the Euclidean norm in R n, respectively, and denote by e i the i-th canonical unit vector. The set of all i-dimensional linear subspaces of R n is denoted by L n i, and for L L n i, L denotes its orthogonal complement. For K K n and L L n i, the orthogonal projection of K onto L is denoted by K L. With lin{u 1,..., u m } we represent the linear hull of the vectors u 1,..., u m and with [u 1, u ] the line segment with end-points u 1, u. Finally, for S R n we denote by conv S the convex hull of S and by bd S its boundary. Moreover, we write relbd S to denote the relative boundary of S, i.e., the boundary of S relative to its affine hull aff S. The diameter, minimal width, circumradius and inradius of a convex body K are denoted by D(K), ω(k), R(K) and r(k), respectively. For more information on these functionals and their properties we refer to [4, pp. 56 59]. If f is a functional on K n depending on the dimension in which a convex body K is embedded, and if K is contained in an affine space A, then we write f(k; A) to stress that f has to be evaluated with respect to the space A. Successive outer and inner radii are defined in the following way. Definition 1.1. For K K n and i = 1,..., n let R i (K) = min R(K L) and r i (K) = max max r( K (x + L); x + L ). L L n i L L n i x L 000 Mathematics Subject Classification. Primary 5A0; Secondary 5A40. Key words and phrases. Successive inner and outer radii, p-sum. Authors are supported by MCI, MTM009-10418, and by Programa de Ayudas a Grupos de Excelencia de la Región de Murcia, Fundación Séneca (Plan Regional de Ciencia y Tecnología 007/010), 04540/GERM/06. 1

BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE It is clear that the outer radii are increasing in i, whereas the inner radii are decreasing in i. Moreover, R i (K) is the smallest radius of a solid cylinder with i-dimensional spherical cross section containing K, and r i (K) is the radius of the greatest i-dimensional ball contained in K. We obviously have R n (K) = R(K), R 1 (K) = ω(k), r n (K) = r(k) and r 1 (K) = D(K). The first systematic study of the successive radii was developed in [1]. For more information on these radii and their relation with other measures, we refer, for instance, to [1,, 5, 10, 1, 13, 14, 18, 19]. From now on we write K0 n to denote the class of convex bodies containing the origin. In [7] Firey introduced the following generalization of the classical Minkowski addition (i.e., vectorial addition). Let p 1 be given. Then, for K, K K0 n, the p-sum of K and K is the unique convex body K + p K for which the support function (1) h(k + p K, ) p = h(k, ) p + h(k, ) p. We recall that h(k, u) = sup { x, u : x K }, u S n 1 (as usual S n 1 denotes the (n 1)-dimensional unit sphere of R n ; see e.g. [0, s. 1.7]). Clearly, when p = 1, formula (1) defines the usual Minkowski sum K + K, and for p = it holds h(k + K, u) = max { h(k, u), h(k, u) }, i.e., K + K = conv(k K ). Moreover, in [7, Theorem 1] it is shown that () K + q K K + p K for all 1 p q. Observe that for the p-sums of sets, except in the case p = 1, the translation invariance is lost. In [15, 16] Lutwak studied p-sums of convex bodies systematically, and developed a theory nowadays known as Brunn-Minkowski-Firey theory. In the last years many important developments of this theory have come out; we mention e.g. [3, 6, 17] and the references inside. In this point we would like to notice that usually p-sums are defined for convex bodies containing the origin as a relative interior point, since this condition is needed for the development of the Brunn-Minkowski-Firey theory; however, regarding the functionals we are working on, this condition can be withdrawn, and thus we allow the origin to lie on the boundary of the convex bodies. The behavior of the diameter, minimal width, circumradius and inradius with respect to the Minkowski sum is well known (see e.g. [0, p. 4]), namely, (3) 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 ), and in [8] we obtained the corresponding relations for the more general successive inner and outer radii. Here we are interested in generalizing those

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES 3 results to the p-sums of convex bodies. Regarding outer radii we prove the following theorem. Theorem 1.1. Let K, K K n 0 (4) and p 1. Then p 1 p R 1 (K + p K ) R 1 (K) + R 1 (K ) for all p 1, 3p p R i (K + p K ) R i (K) + R i (K ) for 1 p, i =,..., n, R i (K + p K ) max { R i (K), R i (K ) } for p, i =,..., n. All inequalities are best possible. Moreover, R n (K + p K ) R n (K) + R n (K ) and there exists no constant c > 0 such that c R i (K + p K ) R i (K) + R i (K ) for i = 1,..., n 1 (see Proposition.1). The above result becomes [8, Theorem 1.1] when p = 1. In the case of the successive inner radii, the result is the following. Theorem 1.. Let K, K K n 0 (5) and p 1. Then p 1 p r n (K + p K ) r n (K) + r n (K ) for all p 1, 3p p r i (K + p K ) r i (K) + r i (K ) for 1 p, i = 1,..., n 1, r i (K + p K ) max { r i (K), r i (K ) } for p, i = 1,..., n 1. All inequalities are best possible. Moreover, r 1 (K+ p K ) r 1 (K)+r 1 (K ) and there exists no constant c > 0 such that c r i (K + p K ) r i (K) + r i (K ) for i =,..., n (see Proposition.). The above result becomes [8, Theorem 1.] when p = 1. We notice that in both Theorems 1.1 and 1., the last two inequalities are valid for all p 1. We point out that the distinction depending on the range of p is needed for the sharpness. In Section we present the proofs of the main theorems, as well as some consequences and remarks. Finally, Section 3 is devoted to study a particular case for which the bounds in Theorems 1.1 and 1. can be improved, the so called p-difference body of a convex set.. Proofs of the main results For p 1 let Bn p be the unit p-ball associated to the p-norm p, i.e., ( n ) 1/p Bn p = x = (x 1,..., x n ) R n : x p = x i p 1, with x = max{ x i : i = 1,..., n}. For the sake of brevity, we will write B n = B n to denote the n-dimensional Euclidean unit ball. Moreover, for L L n i, we will write B i,l = B n L and B p i,l = Bp n L. On the one hand, for any 1 p q, it is a direct consequence of Hölder s inequality for q/p (see e.g. [11, p. 15]) that q p n 1/p 1/q q, which is equivalent to the inclusions B p n B q n n (q p)/(qp) B p n. On the other hand, i=1

4 BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE it is known (see e.g. [9]) that [ e 1, e 1 ] + p + p [ e n, e n ] = Bn q for q 1 such that 1/p + 1/q = 1. Therefore we get, in particular, that { n ( p)/(p) B n for 1 p, (6) [ e 1, e 1 ] + p + p [ e n, e n ] B n for p. We start by proving the lower bound for the outer radii R i (K + p K ) of the p-sum of two convex bodies in terms of the corresponding radii. Proof of Theorem 1.1. In [7, Theorem 1] it is shown that (7) 1 (p 1)/p (K + K ) K + p K K + K. Therefore (p 1)/p R i (K + p K ) R i (K + K ) for all i = 1,..., n, and by [8, Theorem 1.1] we get (p 1)/p R 1 (K + p K ) R 1 (K + K ) R 1 (K) + R 1 (K ), (p 1)/p R i (K + p K ) R i (K + K ) 1 ( Ri (K) + R i (K ) ), i =,..., n, which gives the two first inequalities in (4). Notice also that it always holds K, K K + p K, which leads to R i (K + p K ) max { R i (K), R i (K ) }. Since for any non-negative real numbers a, b 0 it holds that if p then max{a, b} 1/ (3p )/(p) (a + b), the third inequality in (4) is obtained. So it remains to be shown that the three inequalities are best possible. For the first one, let K = K. Then K + p K = 1/p K and thus (p 1)/p R 1 (K + p K) = (p 1)/p 1/p R 1 (K) = R 1 (K) + R 1 (K). Next we fix i {,..., n} and consider the convex bodies n n K = [ e 1, e 1 ] + [ e k, e k ], K = [ e, e ] + [ e k, e k ], k=i+1 k=i+1 i.e., the 0-symmetric (n i + 1)-cubes with edges parallel to the coordinate axes and length, of the subspaces L j = lin{e j, e i+1,..., e n } L n n i+1, j = 1,. Here for i = n we are just taking K = [ e 1, e 1 ], K = [ e, e ]. Clearly R(K L), R(K L) 1 for all L L n i. Moreover, R( K lin{e 1,..., e i } ) = R ( K lin{e 1,..., e i } ) = 1, which shows that R i (K) = R i (K ) = 1. Let 1 p, and we compute R i (K + p K ). Let L L n i. On the one hand, since dim L j L (n i + 1) + i n = 1, there exist x K L and x K L with, x 1, and because of the central symmetry, we may assume that x, x 0. Then, since (x + x )/ (p 1)/p (K + p K ) L (see (7)), we get (x + x ( ) x + x 1/ ) (p 1)/p (p 1)/p 1/ (p 1)/p = ( p)/(p),

and thus, ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES 5 (8) R ( (K + p K ) L ) R ( (K + p K ) L ) ( p)/(p) for all L L n i. On the other hand, since the orthogonal projection of the p-sum of two convex bodies onto any lower dimensional linear subspace is the p-sum of the projections (see [7, pp. 1 ]), and using (6), we get (K + p K ) lin{e 1,..., e i } = K lin{e 1,..., e i } + p K lin{e 1,..., e i } = K lin{e 1 } + p K lin{e } = [ e 1, e 1 ] + p [ e, e ] ( p)/(p) B,lin{e1,e }, which gives R ( (K + p K ) lin{e 1,..., e i } ) ( p)/(p). Then, together with (8) we get R ( (K + p K ) lin{e 1,..., e i } ) = ( p)/(p), and moreover, R i (K + p K ) = min R ( (K + p K ) L ) = ( p)/(p) 1 = (1 + 1) L L n i (3p )/(p) 1 ( = Ri (3p )/(p) (K) + R i (K ) ), as required. Now let p. First notice that R i (K + p K ) R i (K), R i (K ) = 1. With an analogous argument as before, but using (6) when p, we get that R ( (K + p K ) lin{e 1,..., e i } ) R ( B,lin{e1,e }) = 1. Both inequalities give R i (K + p K ) = 1 = R i (K), R i (K ). Regarding a reverse inequality, in the case of the circumradius we easily get, using (7) and (3), that R(K + p K ) R(K + K ) R(K) + R(K ). However, there is no chance to get a reverse inequality for all outer radii, as next proposition shows. Proposition.1. Let K, K K n 0 and p 1. Then R n (K + p K ) R n (K) + R n (K ). However, for any i = 1,..., n 1, there exists no constant c > 0 such that c R i (K + p K ) R i (K) + R i (K ). Proof. We already know that R n (K + p K ) R n (K) + R n (K ). Notice that the inequality is tight, because the circumradius is a continuous functional and equality is attained when K = {0}. In order to show the non-existence of a reverse inequality, i = 1,..., n 1, we take the convex bodies n i K = [ e n i+1, e n i+1 ] and K = [ e k, e k ]. k=1

6 BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE On the one hand, notice that K lin{e n i, e n i+,..., e n } = K lin{e n i+1,..., e n } = {0}, and hence both R i (K) = R i (K ) = 0, i.e., R i (K) + R i (K ) = 0. On the other hand, K + p K 1 (p 1)/p (K + K ) = n i+1 1 (p 1)/p j=1 [ e j, e j ], i.e., K + p K contains an (n i + 1)-dimensional convex body, which implies that dim ( (K + p K ) L ) 1 for all L L n i. Then, R i(k + p K ) > 0. Hence we conclude that there exists no constant c > 0 such that c R i (K + p K ) R i (K) + R i (K ) for any i = 1,..., n 1. Remark.1. Notice that the upper bound for the circumradius of the p-sum of two convex bodies K, K K0 n, p 1, can be improved in the particular case when the circumcenter of both K, K lies in the origin. In this case, h(k + p K, u) = ( h(k, u) p + h(k, u) p) 1/p ( R(K) p + R(K ) p) 1/p for all u S n 1, which implies, in particular, that the circumradius of the p-sum of K, K is not greater than the p-sum of the circumradii, If K = K equality holds. R(K + p K ) ( R(K) p + R(K ) p) 1/p. Now we get the corresponding bounds for the inner radii r i (K + p K ) by proving Theorem 1.. Proof of Theorem 1.. By (7) we have (p 1)/p r i (K + p K ) r i (K + K ) for all i = 1,..., n, and applying [8, Theorem 1.] we get (p 1)/p r n (K + p K ) r n (K + K ) r n (K) + r n (K ), (p 1)/p r i (K + p K ) r i (K + K ) 1 ( ri (K) + r i (K ) ), i = 1,..., n 1, which gives the two first inequalities in (5). Again, since K, K K + p K, then r i (K + p K ) max { r i (K), r i (K ) }, which leads to the third inequality in (5). So, we have to show that these inequalities are best possible. For the first one, with K = K we get (p 1)/p r n (K + p K) = (p 1)/p 1/p r n (K) = r n (K) + r n (K). Next we fix i {1,..., n 1} and consider the following convex bodies: for j = i n if i n, and j = 0 otherwise, take the i-dimensional unit balls K = B i,l and K = B i,l of the i-dimensional linear subspaces L = lin{e 1,..., e j, e j+1,..., e i }, L = lin{e 1,..., e j, e i+1,..., e i j };

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES 7 here, for j = 0 we are taking L = lin{e 1,..., e i } and L = lin{e i+1,..., e i }. Clearly, r i (B i,l ) = r i (B i,l ) = 1. Notice that since B i,l, B i,l are 0-symmetric, then B i,l + p B i,l is also 0-symmetric, and then r i (B i,l + p B i,l ) = max r ( (B i,l + p B i,l ) L; L ). L L n i Let 1 p. We are going to show that (9) r ( (B i,l + p B i,l ) L; L ) ( p)/(p) for all L L n i, which will imply that ( p)/(p) 1 = (3p )/(p) (1 + 1) = 1 ( ri (3p )/(p) (B i,l ) + r i (B i,l ) ) r i (B i,l + p B i,l ) p p, i.e., r i (B i,l + p B i,l ) = 1/ (3p )/(p)( r i (B i,l ) + r i (B i,l ) ), as required. Let L = lin{e j+1,..., e n }. If i < n then L = R n and thus, for all L L n i, it holds dim( L L ) = dim L = i 1; analogously, if i n, then dim( L L ) = dim L + dim L dim( L + L ) = i + n j dim( L + L ) i + n j n = n i 1. Therefore, since dim(b i,l + p B i,l ) = n, then, for all L L n i, we can always find z relbd(b i,l + p B i,l ) L L, z 0. Moreover, notice that, in particular, z lin{e j+1,..., e i j }, and so it can be expressed in the form z = x + x lin{e j+1,..., e i } + lin{e i+1,..., e i j } = L L + L L ; observe that x, x lie in orthogonal subspaces. Writing u = z/ z, we have z = z, u h ( B i,l + p B i,l, u ) ( = h (B i,l, u) p + h ( B i,l, u ) ) p 1/p, and since h(b i,l, u) = sup y, u = 1 sup y, x = 1 y B i,l z y B i,l z ([ = h x ] ) x,, u x, x and analogously h(b i,l, u) = h ( [ x / x, x / x ], u ), we obtain that ( ([ z h x ] x p,, u) + h ([ x x ] ) ) p 1/p x, x, u ( [ = h x ] x, + p [ x x ] ) ( ) x, x, u = h B p/(p 1),lin{x,x }, u ( ) R B p/(p 1),lin{x,x } ( p)/(p) R ( ) B,lin{x,x } = ( p)/(p)

8 BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE by (6). This implies r ( (B i,l + p B i,l ) L; L ) ( p)/(p), showing (9) and concluding the proof of the case 1( p. ( ) ( Now let p. Notice that r i Bi,L + p B i,l ) ri Bi,L, ri Bi,L ) = 1. So, it suffices to show that r ( (B i,l + p B i,l ) L; L ) 1 for all L L n i. With an analogous argument as before, but using (6) when p, we get that there exists z relbd(b i,l + p B i,l ) L L such that ( ) z R B p/(p 1),lin{x,x } R ( B,lin{x,x }) = 1. It shows that r ( (B i,l + p B i,l ) L; L ) 1 and concludes the proof. We deal again with the possible existence of a reverse inequality. In the case of the diameter we easily get, using (7) and (3), that D(K + p K ) D(K + K ) D(K) + D(K ). However, there is no chance to get a reverse inequality for all inner radii, as next proposition shows. Proposition.. Let K, K K n 0 and p 1. Then r 1 (K + p K ) r 1 (K) + r 1 (K ). However, for any i =,..., n, there exists no constant c > 0 such that c r i (K + p K ) r i (K) + r i (K ). Proof. We already know that r 1 (K + p K ) r 1 (K)+r 1 (K ). Notice that the inequality is tight, since the diameter is a continuous functional and equality is attained when K = {0}. In order to show the non-existence of a reverse inequality, i =,..., n, we take the convex bodies i K = [ e 1, e 1 ] and K = [ e k, e k ]. Clearly r i (K) = r i (K ) = 0, because they have dimensions dim K = 1 and dim K = i 1. However, ) k= r i (K+ p K ) (p 1)/p r i (K+K ) = (p 1)/p r i ( i k=1[ e k, e k ] = (p 1)/p, which shows that there exists no constant c > 0 such that c r i (K + p K ) r i (K) + r i (K ) for any i =,..., n. 3. On the p-difference body of a convex set The difference body of a convex body K K n is defined as the Minkowski addition K K := K + ( K) which, in particular, is a 0-symmetric body. Thus, this operation defines the so called central symmetral of a convex body K, just taking K 0 = (K K)/ (see e.g. [4, p. 79] for properties of the central symmetrization).

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES 9 Similarly, the p-difference body can be defined. For K K0 n, we take the p-sum K p K := K + p ( K), which is also 0-symmetric: h(k p K, u) p = h(k, u) p + h( K, u) p = h( K, u) p + h(k, u) p = h(k p K, u) p. In [3], a sharp Rogers-Shephard inequality for the p-difference body of a planar convex body was obtained, i.e., the best (upper) bound for the volume of the set K p K in terms of the volume of the original body K. Here we are interested in obtaining upper and lower bounds for the in- and outer radii of the p-difference body K p K in terms of the ones of K. In [8, Proposition 4.] we already studied the behavior of the radii regarding the usual difference body: for all i = 1,..., n, (10) i + 1 R i (K) R i (K K) R i (K), i r i (K) r i (K K) < (i + 1)r i (K). Next proposition extends the above results to all radii, showing moreover that the bounds in (4) and (5) can be improved and that, in this particular case, there are non-trivial reverse inequalities (cf. Propositions.1 and.). Proposition 3.1. Let K K0 n. Then for all i = 1,..., n and all p 1, (11) If p, 1/p 1/ i+1 i } If p, max { 1/p 1/ i+1 i, 1 R i(k) R i (K p K) R i (K), (1) 1/p r i (K) r i (K p K) < (i + 1)r i (K). The upper bound and the lower bound when 1 p in (11) are best possible. Lower bound in (1) is best possible. Proof. By (7) we have, for all i = 1,..., n, that 1 (p 1)/p R i(k K) R i (K p K) R i (K K), and analogously for the inner radii r i. Then applying (10) we directly get (11) and (1). We observe that if p, for all i and most of the values of p, it holds 1 > 1/p 1/ (i + 1)/i. So we deal with the sharpness of the inequalities, starting with the left hand side in (1). In this case, just notice that if K is a 0-symmetric convex body then K = K and hence r i (K p K) = r i (K + p K) = 1/p r i (K). Next we study the right hand side in (11). We fix i {1,..., n} and consider the convex body K = [0, e 1 ]+ n j=i+1 [ e j, e j ], for which it clearly holds R i (K) = R ( K lin{e 1,..., e i } ) = R ( [0, e 1 ] ) = 1/; here, if i = n we are taking K = [0, e 1 ]. Now, on the one hand notice that (K p K) lin{e 1,..., e i } = [0, e 1 ] + p [ e 1, 0] and that, by (), [ e 1, e 1 ] = conv ( [0, e 1 ] [ e 1, 0] ) [0, e 1 ] + p [ e 1, 0] [0, e 1 ] + [ e 1, 0] = [ e 1, e 1 ],

10 BERNARDO GONZÁLEZ AND MARÍA A. HERNÁNDEZ CIFRE i.e., (K p K) lin{e 1,..., e i } = [0, e 1 ] + p [ e 1, 0] = [ e 1, e 1 ]. On the other hand we observe that conv ( K ( K) ) = C n i+1 is the (n i+1)-dimensional cube of edge-length contained in lin{e 1, e i+1,..., e n } and thus, by (), we get that for all L L n i R ( (K p K) L ) R ( conv ( K ( K) ) L ) = R(C n i+1 L) R ( C n i+1 lin{e 1,..., e i } ) = R ( [ e 1, e 1 ] ) = R ( (K p K) lin{e 1,..., e i } ). Therefore, R i (K p K) = R ( (K p K) lin{e 1,..., e i } ) = 1 = R i (K). Finally we consider the equality case for the left hand side in (11) when 1 p. If i = n, let S n be the n-simplex, embedded in R n+1, lying in the hyperplane S n = conv { x = (x 1,..., x n+1 ) R n+1 : n+1 j=1 x j = 0 { p k : p kk = n n + 1, p kj = 1 n + 1 }, given by } for j k, k = 1,..., n + 1. Since S n p S n is a 0-symmetric convex body, then n+1 R n (S n p S n ) = max h(s n p S n, u) : u = 1 and u j = 0. Let u R n+1 with u = 1 and n+1 j=1 u j = 0. Recall that the value of the support function of a convex body at any vector is attained in an extreme point (cf. e.g. [11, Theorem 5.6]), so, in order to compute h(s n, u) it suffices to consider the vertices of S n. Since p k, u = n n + 1 u k 1 u j = u k, n + 1 then h(s n, u) = max { p k, u : k = 1,..., n + 1 } = max{u 1,..., u n+1 }. For the sake of brevity, we assume that u 1 u n+1. Then h(s n p S n, u) p = h(s n, u) p + h( S n, u) p = h(s n, u) p + h(s n, u) p = u p 1 + ( u n+1) p. By elementary but tedious calculations one can show that the maximum of the function u p 1 + ( u n+1) p, 1 p, under the conditions u = 1 and n+1 j=1 u j = 0, is attained in the point ( 1/, 0,..., 0, 1/ ). Therefore, ( 1 R n (S n p S n ) = p/ + 1 ) 1/p p/ = 1/p 1/. Since R n (S n ) = n/(n + 1), then we get the required equality: R n (S n p S n ) = 1/p 1/ = 1/p 1/ n + 1 n R n(s n ). If i < n, we take the i-dimensional simplex S i and consider the convex body K = S i + MC n i, where C n i (aff S i ) represents the (n i)-dimensional j k j=1

ON SUCCESSIVE RADII AND p-sums OF CONVEX BODIES 11 unit cube and M > 0 is sufficiently large such that R i (K p K) = R(S i p S i ) and R i (K) = R(S i ). The above argument gives the result. Acknowledgement. The authors would like to thank E. Saorín, who called our attention in this problem, for enlightening discussions. References [1] U. Betke, M. Henk, Estimating sizes of a convex body by successive diameters and widths, Mathematika 39 () (199), 47 57. [] U. Betke, M. Henk, A generalization of Steinhagen s theorem, Abh. Math. Sem. Univ. Hamburg 63 (1993), 165-176. [3] C. Bianchini, A. Colesanti, A sharp Rogers and Shephard inequality for the p- difference body of planar convex bodies, Proc. Amer. Math. Soc. 136 (7) (008), 575-58. [4] T. Bonnesen and W. Fenchel, Theorie der konvexen Körper. Springer, Berlin, 1934, 1974. English translation: Theory of convex bodies. Edited by L. Boron, C. Christenson and B. Smith. BCS Associates, Moscow, ID, 1987. [5] R. Brandenberg, Radii of regular polytopes, Discrete Comput. Geom. 33 (1) (005), 43 55. [6] S. Campi, P. Gronchi, Volume inequalities for L p -zonotopes, Mathematika 53 (1) (006), 71-80 (007). [7] Wm. J. Firey, p-means of convex bodies, Math. Scand. 10 (196), 17-4. [8] B. González, M. A. Hernández Cifre, Successive radii and Minkowski addition, to appear in Monatsh. Math. (01). [9] Y. Gordon, M. Junge, Volume formulas in L p -spaces, Positivity 1 (1) (1997), 7-43. [10] P. Gritzmann, V. Klee, Inner and outer j-radii of convex bodies in finite-dimensional normed spaces, Discrete Comput. Geom. 7 (199), 55-80. [11] P. M. Gruber, Convex and Discrete Geometry. Springer, Berlin Heidelberg, 007. [1] M. Henk, A generalization of Jung s theorem, Geom. Dedicata 4 (199), 35 40. [13] M. Henk, M. A. Hernández Cifre, Intrinsic volumes and successive radii, J. Math. Anal. Appl. 343 () (008), 733 74. [14] M. Henk, M. A. Hernández Cifre, Successive minima and radii, Canad. Math. Bull. 5 (3) (009), 380 387. [15] E. Lutwak, The Brunn-Minkowski-Firey theory, I, J. Differential Geom. 38 (1) (1993), 131-150. [16] E. Lutwak, The Brunn-Minkowski-Firey theory, II, Adv. Math. 118 () (1996), 44-94. [17] E. Lutwak, D. Yang, G. Zhang, On the L p-minkowski problem, Trans. Amer. Math. Soc. 356 (11) (004), 4359-4370. [18] G. Ya. Perel man, On the k-radii of a convex body, (Russian) Sibirsk. Mat. Zh. 8 (4) (1987), 185 186. English translation: Siberian Math. J. 8 (4) (1987), 665 666. [19] S. V. Puhov, Inequalities for the Kolmogorov and Bernšteĭn widths in Hilbert space, (Russian) Mat. Zametki 5 (4) (1979), 619 68, 637. English translation: Math. Notes 5 (4) (1979), 30 36. [0] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, Cambridge University Press, Cambridge, 1993. Departamento de Matemáticas, Universidad de Murcia, Campus de Espinardo, 30100-Murcia, Spain E-mail address: bgmerino@um.es E-mail address: mhcifre@um.es