INRADII OF SIMPLICES

Transcription

1 INRADII OF SIMPLICES ULRICH BETKE, MARTIN HENK, AND LYDIA TSINTSIFA Abstract. We stuy the following generalization of the inraius: For a convex boy K in the -imensional Eucliean space an a linear k-plane L we efine the inraius of K with respect to L by r L(K) = max{r(k; x + L) : x E }, where r(k; x + L) enotes the orinary inraius of K (x + L) with respect to the affine plane x + L. We show how to etermine r L(P ) for polytopes an use the result to estimate min{r L(T r ) : L is a k-plane} for the regular -simplex T r. These estimates are optimal for all k in infinitely many imensions an for certain k in the remaining imensions. 1. Introuction We enote by E the -imensional Eucliean space equippe with the norm an inner prouct,. The space of all compact convex boies is enote by K an B K enotes the -imensional unit ball. By L k we enote the space of all k imensional linear subspaces of E an for L L k, M E we write M L for the orthogonal projection of M onto L. The orthogonal complement of L L k is enote by L. The inraius r(k), the circumraius R(K), the iameter D(K) an the minimal with (K) are classical funamental functionals of a convex boy K K. For a etaile escription we refer to [BF34]. More information on the boy can be obtaine if we link each pair of the functionals by a series of 2 intermeiate functionals. It turns out that these functionals allow generalizations an analogues of many classical results. A first systematic stuy of these series can be foun in [Hen91] an for recent work see e.g. [Bal92], [Hen92], [BH92], [BH93]. Some of these intermeiate functionals are well known functionals in approximation theory, calle Bernstein an Kolmogorov iameters, (cf. e.g. [Pin85], [Puk79]) an they are also of interest for computational aspects of convex boies (cf. e.g. [GK93], [GHK90], [BH93]). Here we stuy a series linking the inraius an minimal with. There are two natural series of this kin. For the efinition we nee some more notation: For a k-imensional plane L an a set M E we write r(m; L) for the inraius of M with respect to the space L, i.e., r(m; L) is the raius of the largest k-imensional ball containe in M L. A ball with raius r(m; L) containe in M L is calle an inball. For x E the translate {l + x : l L} is enote by L(x). The work of the secon author is supporte by the Gerhar-Hess Forschungs- Förerpreis of the German Science Founation (DFG) aware to Günter M. Ziegler (Zi 475/1-1). The paper contains some material of the Ph.D. thesis of the thir author. We wish to thank the referees for helpful comments an suggestions. 1

2 2 ULRICH BETKE, MARTIN HENK, AND LYDIA TSINTSIFA Definition 1. For K K an 1 k let rk π (K) = min r(k L; L) an r L L k σ (K) = min k Obviously, we have L L x E k max r(k; L(x)). r(k) = r π (K) rπ 1 (K) rπ 1 (K) = (K)/2, r(k) = r σ (K) rσ 1 (K) rσ 1 (K) = (K)/2. Furthermore, we have rk π(k) rσ k (K) an there exist convex boies K K, 3, with rk π(k) > rσ k (K) [Tsi96]. For a -simplex T K an explicit formula for rk π(t ) was given in [BH93]. Apparently the situation for the rk σ (K) is much more complicate, as we have a twofol optimization. Here we give formulae for the inner optimization an then we use these formulae to compute the rk σ(t r ) for the regular -simplex Tr. For the analogous series rσ(k) k = max L L max k x E r(k; L(x)) this was one by K. Ball (cf. [Bal92]). There it turne out that the rσ(t k r ) are the orinary inraii of k imensional faces. Unfortunately, the problem to characterize the optimal planes for rk σ(t r ) is much more intractable. In particular, we will see that the value of rk σ(t r ) epens on the parity of the imension. Base on a formula for rk σ(t ) for arbitrary simplices T we will give estimates for rk σ(t r ) which are optimal for infinitely many imensions an for certain k in the remaining imensions but o not hol in general. The first case that the estimate is not tight occurs for even an k = 1 or k = 1. However, in this instance we have again an exact formula. In more etail our results are as follows: For L L k an K K let (1) r Lk (K) = max x E r(k; L k(x)). For polytopes we have Theorem 1. Let P E be a -polytope an L k L k. Then r L k (P ) is attaine for a plane L k (x) such that an inball of P L k (x) touches at least + 1 facets of P. By Theorem 1 the computation of rk σ (P ) is reuce to the computation of rk σ(t ) for finitely many simplices T P. For a -simplex T Theorem 1 yiels an effective formula: Let u 0,..., u +1 be the outer normal vectors of the facets of T such that u i is equal to the area of the corresponing facet. u 0,..., u are calle the facets vectors of T. With this notation we have Theorem 2. Let T E be a -simplex with facets vectors u 0,..., u an L k L k. Then r Lk (T ) = r(t ) ui ui L k. To state our last Theorem about the inraii of a regular simplex we nee Haamar numbers:

3 INRADII OF SIMPLICES 3 Definition 2. A p p matrix H = (h ij ) is calle a Haamar matrix if h ij = ±1 for all i, j an HH T = pi p where H T enotes the transpose of H an I p enotes the unit matrix. A number p, for which a Haamar matrix exists, is calle a Haamar number. It is well known that p = 2 m is a Haamar number for all m N an it is conjecture that p = 4m is a Haamar number for all m N. The conjecture has been verifie for m 106 an in many other cases as well (cf. [Miy91]). Theorem 3. For a regular -simplex T r with r(t r ) = 1 one has r σ k (T r ) k for k = 1,...,. Equality hols (at least) in the following cases: (1) for k = m if m + 1 ivies + 1, (2) for all k if + 1 is a Haamar number, (3) for k = 2 an for o, k = 3. If equality hols for the pair (k, ) then we also have equality in the cases ( k, ), (k, n( + 1) 1), (nk, n( + 1) 1), n N. For even it hols r σ 1 (T r ) = , r σ 1 (T r ) = It is certainly of interest to compare rk π(t r ) an rk σ(t r ). This shows a strange behaviour of these numbers. If we take the imension such that +1 is a Haamar number we see (cf. (10)) that rk π(t r ) an rk σ(t r ) coincie if an only if k + 1 ivies + 1. We observe that our result implies many examples for rk π(k) rσ k (K) an the relation of rπ k (T r ) an rk σ(t r ) epens rather on the number-theoretical properties of k an than on their sizes. In section 2 we prove Theorem 1 an Theorem 2 an in section 3 we stuy rk σ(t r ). In orer to simplify the proof of Theorem 3 we split it in several lemmas. In the last section we point out a relation between the inraii of arbitrary convex boies an simplices. Finally, we remark that the problem to etermine rk σ (K) may be consiere as a special case of the more general problem to obtain information about a convex boy via inscribe largest convex boies. For questions of this type we refer to [GKL95] an [HKL95]. 2. Optimal planes an Inraii of simplices We start with the proof of the characterization of the planes for which r Lk (P ) is attaine (cf. (1)). Proof of Theorem 1. Let w 1,..., w m be the outwar unit normal vectors of the facets F 1,..., F m of the polytope P an let b 1,..., b m R such that P = {x E : w i, x b i, 1 i m}. Moreover, let v 1,..., v be an orthonormal basis of E such that L k is generate by v 1,..., v k. For abbreviation we write ŵ i instea of w i L k. Now,

4 4 ULRICH BETKE, MARTIN HENK, AND LYDIA TSINTSIFA for x E an a positive number r we consier the functions f i (x, r) = w i, x + r ŵ i, 1 i m. First, we prove the following relation (2) f i (x, r) b i, 1 i m, x + r(b L k ) P L k (x). In orer to show we may assume ŵ i 0. In this case we have x + rŵ i / ŵ i P an hence (3) f i (x, r) = w i, x + rŵ i / ŵ i b i. For the reverse irection we note that x P L k (x) an P L k (x) is a nonempty polytope. W.l.o.g. let F 1,..., F n be the facets of P such that F i L k (x), 1 i n, are the facets of the polytope P L k (x). Then we have ŵ i 0 an the outwar unit normal vector of the facet F i L k (x) with respect to P L k (x) is given by ŵ i / ŵ i. Hence it suffices to show x + rŵ i / ŵ i P, 1 i n, what is equivalent to f i (x, r) b i, 1 i n, an (2) is prove. Now, suppose f i (x, r) b i, 1 i m. By our last argument we have further ( ) (4) f i (x, r) = b i x + r(b L k ) F i. In orer to prove the theorem we choose an x E such that the inball x+ r Lk (P )(B L k ) P touches a maximal number of facets of P. W.l.o.g. let F 1,..., F l be these facets. By (2), (4) we have f i (x, r Lk (P )) = b i, 1 i l, an f i (x, r Lk (P )) < b i, l + 1 i m. Now, we assume l < + 1 an istinguish two cases. i) The vectors w 1,..., w l are linearly inepenent. Then we can fin a z E with w i, z < 0, 1 i l. Hence for sufficiently small ɛ > 0 we get f i (x + ɛz, r Lk (P )) < b i, 1 i m, which contraicts the efinition of r Lk (P ) (cf. (2)). ii) The vectors w 1,..., w l are linearly epenent. Then there exists a z E \{0} with w i, z = 0, 1 i l. So we have f i (x + ɛz, r Lk (P )) = b i, 1 i l, for ɛ R. However, since P is boune there exists a ɛ > 0 such that f i (x + ɛz, r Lk (P )) b i, 1 i m, but at least l + 1 inequalities are satisfie. This contraicts the choice of the point x an the proof is complete. Applie to a -imensional simplex T the last theorem says that all facets are touche by an optimal k-imensional inball an, in particular, this inball is uniquely etermine. This property is the key tool for the proof of Theorem 2. Proof of Theorem 2. W.l.o.g. let r(t ) = 1 an let T be given by T = {x E : u i / u i, x 1}. Let v 1,..., v be an orthonormal basis of E such that L k is generate by v 1,..., v k. Furthermore, let x E such that r Lk (T ) = r(t ; L k (x)) an x

5 INRADII OF SIMPLICES 5 is the center of the inball of T L k (x). By Theorem 1 x+r Lk (T )(B L k ) touches all facets of T an we have (cf. (4)) Summing up gives u i, x + r Lk (T ) u i, x + r Lk (T ) u i L k = u i. u i L k = u i. Now, it is well known that ui = 0 (cf. [BF34]) an thus we obtain the formula. So, by Theorem 2 the problem to etermine rk σ(t ) for a -simplex T is reuce to the etermination of { } (5) min ui ui L k : L k L k. In other wors, we have to fin a k-imensional plane L k such that the sum ui L k becomes maximal. For regular simplices an certain cases of k an we can explicitly compute the exact maximum. 3. Regular simplices Let Tr be a regular -simplex with r(tr ) = 1 an let 0 be the center of the -imensional inball. Hence, T r = {x E : w i, x 1, 1 i }, where w i are the outwar unit normal vectors of the facets of T r. In this case Theorem 2 gives (6) r σ k (T r ) = Lemma max{ wi L k : L k L k }. (7) r σ k (T r ) /k an equality hols if an only if there exists a L k L k such that w 0 L k = w 1 L k = = w L k = k/. Proof. For the proof we use the fact that for v E, v = 1, it hols (cf. [Bal92], [Tsi96]) (8) w i, v 2 = + 1. Let L k L k be an arbitrary k-plane an let v1,..., v be an orthonormal basis of E such that L k is spanne by v 1,..., v k. Then we have w i L k 2 = k wi, v j 2 an by (8) w i L k 2 = k w i, v j 2 = k + 1.

6 6 ULRICH BETKE, MARTIN HENK, AND LYDIA TSINTSIFA Application of the Cauchy-Schwarz inequality yiels (9) ( w i L k ) 2 ( + 1) w i L k 2 = k ( + 1)2. Together with (6) this shows (7). Furthermore, (9) is satisfie with equality if an only if all w i L k have the same length. In orer to boun r σ k (T r ) from above we use the explicit formula for r π k (T r ) given in [BH93] (Lemma 3.2) (10) rk σ (T r ) rk π (T r ) ( = m l ( + 1 l ) ( + 1) 2 + (k + 1 m) ) 1 l ( + 1 l ) ( + 1) 2, where l = ( + 1)/(k + 1), + 1 m (mo k + 1), m {0,..., k}, an x ( x ) enotes the smallest (largest) integer ( ) x. Since l ( + 1)/2 we obtain ( ) 1 rk σ (T r l ( + 1 l ) ) (k + 1) ( + 1) 2 an writing l = ( + 1)/(k + 1) µ( + 1)/(k + 1) gives rk σ (T r ) k k (1 µ)(k + µ). Thus, by Lemma 1 we have Lemma 2. r σ k (T r ) = k if k + 1 ivies + 1. Moreover, since the parameter µ is boune from above by min{k, k}/( + 1) it is not har to see that rk σ (T r ) k In general we can not expect to fin a k-plane such that all the projections of w i onto that plane have equal lengths as require by Lemma 1. However, for certain constellations of k an we can construct such planes. To this en we use the following approach: Let L k be a k-plane given by an orthogonal basis v 1,..., v k, v i = 1. Each v i can be uniquely represente as v i = x i jw j with j=0 x i j = 0, 1 i k. j=0

7 INRADII OF SIMPLICES 7 Using the ientity w m, w n = 1/, m n, we obtain for the coorinates x i j the relations v i = 1 v m, v n = 0 (x i j) 2 = /( + 1), 1 i k, j=0 x m j x n j = 0, m n. j=0 Furthermore, we fin for the length of the projection w j L k k w j L k = w j, v i v i = + 1 k (x i j )2. i=1 i=1 In view of Lemma 1 we get with a suitable normalization Lemma 3. r σ k (T r ) = /k iff there exist x 1,..., x k R +1 with (11) i) x i = 1, 1 i k, ii) x m, x n = 0, 1 m < n k, +1 k iii) x i j = 0, 1 i k, iv) (x i j) 2 = k + 1, 1 j + 1. i=1 An immeiate consequence is Lemma 4. If + 1 is a Haamar number then one has equality in (7) for all k = 1,...,. Proof. Since + 1 is a Haamar number there exist pairwise orthogonal vectors y 0, y 1,..., y R +1 with coorinates yj i { 1, 1}. W.l.o.g. let y 0 = (1, 1,..., 1) T. Then we have +1 yi j = 0, 1 i, an each subset of the vectors x i = ( + 1) 1/2 y i, 1 i, satisfies (11). The vectors x i constructe in the previous proof are of a special type, because all coorinates have the same absolute value. Obviously, for o numbers there o not exist even two vectors of this type satisfying (11). However, as the next lemma shows, in all imensions we can fin two vectors satisfying the conitions of (11). Lemma 5. Equality hols in (7) for k = 2 an for o, k = 3. Proof. First, we stuy the case k = 2. For j = 1,..., + 1 let α j = 2πj , x1 j = + 1 cos α j, x 2 j = + 1 sin α j. In the following we verify the properties i) iv) of (11) for the vectors x 1, x 2. Obviously, (x 1 j )2 + (x 2 j )2 = 2/( + 1), which shows iv). Since cos α j + i sin α j are the complex roots of the equation x +1 1 = 0 we also have iii), i.e.,

8 8 ULRICH BETKE, MARTIN HENK, AND LYDIA TSINTSIFA +1 x1 j = +1 x2 j = 0. Moreover, +1 x 1 jx 2 j = +1 (x 1 j) 2 = cos α j sin α j = 1 +1 sin 2α j, cos 2 α j = 1 +1 (1 + cos 2α j ) = cos 2α j, (x 2 j) 2 =... = sin 2α j. + 1 It remains to show +1 sin 2α j = +1 cos 2α j = 0. Now, if + 1 is o then the numbers cos 2α j + i sin 2α j, 1 j + 1, are the roots of x +1 1 = 0, otherwise cos 2α j + i sin 2α j, 1 j ( + 1)/2, an cos 2α j + i sin 2α j, ( + 1)/2 < j + 1 are the roots of x (+1)/2 1 = 0. For o an k = 3 we choose the coorinates 2 2 x 1 j = + 1 cos 2α j, x 2 j = + 1 sin 2α j, 1 j + 1, { x 3 1/ + 1, j = 1,..., +1 j = 2, 1/ + 1, j = +3 2,..., + 1. This case can be treate completely similar to the case k = 2. The next lemma shows that equality in (7) for some (k, ) implies equality for certain other values of k an. Lemma 6. If equality hols in (7) for the pair (k, ) then we also have equality for in the cases ( k, ), (k, n( + 1) 1), (nk, n( + 1) 1), n N. Proof. Suppose, we have equality for (k, ). By Lemma 1 there exists a k-plane L k such that w i L k = k/, 0 i. Obviously, for the orthogonal complement L k L k of L k we have w i L k 2 = 1 w i L k 2 an hence by Lemma 1 we also have equality for ( k, ). Now, let x 1,..., x k R +1 be vectors satisfying the conitions (11). For an integer n let y i = (x i, x i,..., x i ) R n(+1) be the vector consisting of n copies of x i. With x i = n 1/2 y i we obtain k vectors satisfying (11) with respect to the imension n(+1) 1. In orer to show equality for the pairs (nk, n( + 1) 1), n N, let y i,m R n(+1), 1 i k, 1 m n, be the vectors with coorinates { x y i,m i j = j (m 1)(+1), j {(m 1)( + 1) + 1,..., m( + 1)} 0, otherwise.

9 Now, we are reay for the proof of Theorem 3. INRADII OF SIMPLICES 9 Proof of Theorem 3. By the previous lemmas it remains to stuy the case even an k = 1, 1. Since r1 σ(t r ) is one half of the minimal with of the regular simplex we may euce the value of r1 σ(t r ) from the Theorem of Steinhagen (cf. [Ste22], [BH93]) (12) r(k) (K) 2 { + 2/( + 1), even, 1/, o,, K K, where, for example, equality hols for a regular simplex. Unfortunately, for k = 1 the proof is rather lengthy an teious, an thus we omit it here an we refer to [Tsi96]. 4. Concluing Remarks By Theorem 3 we know the exact values of r σ k (T r ) for 5 an the first unknown value is r σ 3 (T 6 r ). Moreover, since we have equality for k = 2 in all imensions Theorem 3 shows that we also have equality for k = 4 in all o imensions. Next we want to show a relation between the inraii of an arbitrary convex boy an the inraii of simplices. To this en we efine σ(k, ) = inf{r(t )/rk σ (T ) : T is a -simplex}. Theorem 3 implies the upper boun σ(k, ) k/ an we conjecture that σ(k, ) is attaine for the regular simplex, i.e., Conjecture 1. For 1 k it hols σ(k, ) = r(t r )/r σ k (T r ). This conjecture is not only of interest in its own, but it woul also lea to a generalization of the Theorem of Steinhagen (12) as the next lemma shows. Lemma 7. Let K K be a convex boy with nonempty interior. It hols r(k)/rk σ (K) σ(k, ). Proof. It is well known that there exists a m-imensional plane L m an a m- simplex T m L m with K L m T m, r(k) = r(t m ; L m ) an the cyliner P = T m + L m contains K (cf. [BF34]). Now, let Tj P, j N, be a sequence of -simplices such that K Tj + ɛ jb for some ɛ j 0 with lim j ɛ j = 0. By construction we have r(tj ) r(t m ; L m ) = r(k) an rk σ(k) µ jrk σ(t j ) for some µ j 1 with lim j µ j = 1. Altogether we get r(k) 1 r(tj lim inf ) (K) j µ j rk σ(t j σ(k, ). ) r σ k

10 10 ULRICH BETKE, MARTIN HENK, AND LYDIA TSINTSIFA References [Bal92] K. Ball, Ellipsois of Maximal Volume in Convex Boies, Geometriae Deicata 41 (1992), [BH92] U. Betke an M. Henk, Estimating sizes of a convex boy by successive iameters an withs, Mathematika 39 (1992), [BH93] U. Betke an M. Henk, A generalization of Steinhagen s theorem, Abh. Math. Sem. Univ. Hamburg 63 (1993), [BH93] U. Betke an M. Henk, Approximating the volume of convex boies, Discrete Comput. Geom. 10 (1993), [BF34] T. Bonnesen an W. Fenchel, Theorie er konvexen Körper, Springer, Berlin, [GHK90] P. Gritzmann, L. Habsieger an V. Klee, Goo an ba raii of convex polygons, Siam J. Computing 20 (1991), No. 2, [GK93] P. Gritzmann an V. Klee, Computational complexity of inner an outer j raii of convex boies in finite imensional norme spaces, Math. Programming 59 (1993), [GKL95] P. Gritzmann, V. Klee an D.G. Larman, Largest j-simplices in n-polytopes, Discrete Comput. Geom. 13 (1995), No. 3-4, [Hen91] M. Henk, Ungleichungen für sukzessive Minima un verallgemeinerte In un Umkugelraien, Dissertation Universität-GH Siegen, [Hen92] M. Henk, A generalization of Jung s theorem, Geometriae Deicata 42 (1992), [HKL95] M. Huelson, V. Klee an D. Larman, Largest j-simplices in -cubes: Some relatives of the Haamar maximum eterminant problem, Preprint, (1995). [Miy91] M. Miyamoto, Construction of Haamar matrices, J. Combin. Theory Ser. A, 57 (1991), [Pin85] A. Pinkus, n withs in Approximation Theory, Springer, Berlin, [Puk79] S.V. Pukhov, Inequalities between the Kolmogorov an the Bernstein Diameters in a Hilbert Space, Math. Notes 25 (1979), [Ste22] P. Steinhagen, Über ie größte Kugel in einer konvexen Punktmenge, Abh. Math. Sem. Univ. Hamburg 1 (1922), [Tsi96] L. Tsintsifa, Verallgemeinerte Inkugelraien konvexer Körper, Dissertation Universität-GH Siegen, Universität Siegen, Fachbereich Mathematik, Hölerlinstr. 3, Siegen, Germany aress: betke@mathematik.uni-siegen.400.e Technische Universität Berlin, Sekr. MA 6-1, Straße es 17. Juni 136, Berlin, Germany aress: henk@math.tu-berlin.e Universität Siegen, Fachbereich Mathematik, Hölerlinstr. 3, Siegen, Germany