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On Simplexes Inscribed in a Hypersurface

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On Simplexes Inscribed in a Hypersurface ON SIMPLEXES INSCRIBED IN A HYPERSURFACE M. L. Gromov UDC 513.83 The sufficient conditions are obtained for the existence, on a hypersurface M c R n, of k points whose convex hull forms a (k- D-dimensional simplex, homothetic to a given simplex A c R n. In particular, it is shown that if M is a smooth hypersurface, homeomorphic to a sphere, such points will exist for any simplex A c R n. The proofs are based on simple topological considerations. There are six references. w i. Introduction Let A c R n be a rectilinear m-dimensional simplex with vertices 50, 51 ..... 5 m, and q a real num- ber. We shall say that a simplex A can be q-inscribed in a set A c R n if there exists a vector y E R n such that q5 i + y E A for i = 0, 1 ..... m. We shall say that a simplex A can be q-immersed in a closed region c R n if for any nonnegative continuous function f(w}, assigned in the region ~2 and vanishing on its bound- ary, there exists a vector y E R n such that q~i +Y ~ ~2 for i = 0, 1, .... m and / (qs0 + y) = ! (q81 + y) ..... I (q~ + y), or, what amounts to the same, the simplex A c R n c R n+l can be q-inscribed in a graph P c R TM of the function f(w}. THEOREM 1. Let ~ c R n be a closed bounded region such that its boundary M is a Cl-submanifold of the space R n and its Euler characteristic X(~) is nonzero. Let A c R n be a rectilinear simplex. Then: a) there exists a positive q such that the simplex A can be q-inscribed in M; b) there exists a positive p such that for any positive q < p the simplex A can be q-immersed in the region ~. If V c R n is a convex body with boundary W and v E W, we shall denote by s(v; V) the (closed) subset of the unit sphere S n-1 c R n that corresponds to the point v under a spherical mapping (see [117 of the hyper- surface W. The boundary M of a convex body ~2 is said to be smooth with respect to an n-dimensional simplex A c R n if for any point w E M there exists a vertex 6 i of A such that s(w; ~) n s(6i; A) = 4) (let us note that if the boundary M is smooth with respect to every n-dimensional simplex A C R n, then M will be a C~-sub - manifold of the space R n }. THEOREM 2. Let ~2 c R n be a bounded convex region with boundary M that is smooth with respect to an n-dimensional simplex A c R n. Then: a) there exists a positive q such that the simplex A can be q-inscribed in M; b) there exists a positive p such that for any positive q < p the simplex A can be q-immersed in ~2. The proof of Theorem 1 is based on simple and well-known topological results, set forth in w 2. Theorem 2 will be derived from Theorem 1 in w 4. In the remarks at the end of the paper we shall discuss (without proof) the uniqueness of an inscribed simplex and the importance of the restrictions involved in the conditions of Theorems 1 and 2. In the following we shall denote by M + ~, where M c R n and ~E R n, the subset obtained from M by a shift by the vector ~. Leningrad Zhdanov State University. Translated from Matematicheskie Zametki, Vol. 5, No. 1, pp. 81-89, January, 1969. Original article submitted December 25, 1967. 52 w 2. Topological Lemmas LEMMA 1. Let an oriented Cl-submanifold M c R n be the boundary (with allowance for the orienta- tion) of a bounded closed region f2 c R n whose orientation is induced from the space R n. Let t(M) be the absolute value of the degree of the tangential mapping 7 : M --* S n-l, where S n-1 c R n is the unit sphere. Then t (M) = IX (~)1. (1) The proof of formula (1) can be found in Hopf's paper [4] (see also [6]). Remark 1. If n is odd, formula (1) will be equivalent to another well-known formula of Hopf: t t (M) =-~ IX (M)I (see [31). Let Z0, Zt ..... Z k be integral cycles of the space R n such that the intersection of their supports is empty and dim Zo -{- dim Z 1 -t- ... q- dim Z k ~ nk -- t. Let us consider webs H i, i = 0, 1 ..... k, spanned over the cycles Z i in the half-space R n+l = R+ x R n D 0 • R n = R n (R+ being a positive ray). In [5] we can find the definition of the index of intersection i(II 0, 171, .... II k) of webs H i, this index being independent of the choice of the webs. Definition 1. The index of intersection i(110, 111 .... , II k) is called the linking coefficient k(Z0, Z 1..... Z k) of the cycles Z0, Zt ..... Zk. The various definitions of the linking coefficient and the proof of their equivalence for the case of two cycles can be found in [2]. The analysis for a larger number of cycles is entirely similar. In Lemmas 2 and 3 we present without proof the properties of the linking coefficient to be used by us below. The proof of these lemmas is based on the standard use of the properties of the index of intersection [5]~ LEMMA 2. By Z~ c R kn let us denote the image of the cycle Z 0 c R n for a diagonal immersion R ~ --> R '~ X R" X... X R '~, v k times and by ~ c Rim we shall denote the cycle Z1 X Z~ X I I I X Z~ ~ R~ X ~X i i I X R~I Then Ik (Z;, ~)1 = I k (Z,, Zl .... , Zk)l, where k(Z~, ~) is the linking coefficient of the cycles Z~ and ~ in the space R Ira. LEMMA 3. Let M m and N n be closed oriented C2-submanifolds of the space Rm+n+l. By U~ we shall denote a tubular neighborhood of N n. Let M m be contained in the set U[,/\N n. We shall assume that U~ = N n • D m+l (D~TM is a sphere of radius e), and denote by p: V~q -- D m+l the projection onto the com- ponent of the direct product. Let us consider the restriction PM: Mm ~ Dm+t of the projection p. Under the mapping PM the image Mp of the manifold M m does not intersect with the center of the sphere D m+l, so that it is possible to project the image Mp onto the boundary S m of the sphere D m+l from its center. Let 0 : M m ~ S~n be a mapping constructed in this way. Then the degree of the mapping 0 will be equal to the linking coefficient of the cycles, determined by the submanifolds M m and N n c R m+n+t. Let M be a closed oriented (n- D-dimensional Cl-submanifold of the space R n, and let E = (61 ..... en) be a basis in R n. By Mbi (b real) we shall denote the submanifolds M + b~i, where i = 1, 2 ..... n. By 53 a(M; E) we shall denote the largest number b such that for any positive c < b wehave M N Mc, 1 N ... f~ Mc, n = r If such a large number does not exist, we shall write a(M; E) = ~o. It is clear that for a sufficiently small positive b the intersection M N Mb, t f] ... N Mb, n willbe empty, and therefore a(M; E) > 0. Definition 2. The self-linking coefficient ~(M) of a submanifold M c R n is defined as the absolute value of the linking coefficient of cycles, determined (oriented!) by the submanifolds M, Mb, 1..... Mb, n, where 0 < b < a(NI; E). It is clear that the quantity ~(M) depends neither on the choice of the basis E, nor on the choice of a positive b < a(M; E), so that the definition is correct. Proposition 1. The self-linking coefficient ~<(Yl) is equal to the absolute value t (M) of the degree of the tangential mapping T : M -- Sn-1. Proof; Without loss of generality it can be assumed that the submainfold M has smoothness of class C 2, and that the basis E is formed by orthogonal unit vectors ep e2 ..... en. Let us denote by M* c R n2 the image of the manifold M c R n in the case of a diagonal immersion R"• R"•215 • ,~ times and by N c R n2 the direct product M xM:<... • ~ R'~xR~'x...• '~ n times n times By $6R"• x ... x R" I r n d~les we shall denote a vector with components (el, ez ..... en), where ei E R n. Let us consider a tubular neighborhood U~q of the submanifold N c R n2. By selecting the direction of of the normal of the submanifold M c R n, we specify the decomposition U~q = N • D n and the projection p : U~q -- D~.
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