On Properties of a Regular Simplex Inscribed into a Ball

Let $B$ be a Euclidean ball in ${\mathbb R}^n$ and let $C(B)$ be a space of~continuous functions $f:B\to{\mathbb R}$ with the uniform norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|.$ By $\Pi_1\left({\mathbb R}^n\right)$ we mean a set of polynomials of degree $\leq 1$, i.e., a set of linear functions upon ${\mathbb R}^n$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ with the nodes $x^{(j)}\in B$ is defined by the equalities $Pf\left(x^{(j)}\right)= f\left(x^{(j)}\right)$, $j=1,$ $\ldots,$ $ n+1$. The norm of $P$ as an operator from $C(B)$ to $C(B)$ can be calculated by the formula $\|P\|_B=\max_{x\in B}\sum |\lambda_j(x)|.$ Here $\lambda_j$ are the basic Lagrange polynomials corresponding to the $n$-dimensional nondegenerate simplex $S$ with the vertices $x^{(j)}$. Let $P^\prime$ be a projector having the nodes in the vertices \linebreak of a regular simplex inscribed into the ball. We describe the points $y\in B$ with the property $\|P^\prime\|_B=\sum |\lambda_j(y)|$. Also we formulate a geometric conjecture which implies that $\|P^\prime\|_B$ is equal to the minimal norm of an interpolation projector with nodes in $B$. We prove that this conjecture holds true at least for $n=1,2,3,4$. Keywords: regular simplex, ball, linear interpolation, projector, norm


Introduction
Let Ω be a convex body in R n . Denote by C(Ω) a space of continuous functions f : Ω → R with the uniform norm f C(Ω) := max x∈Qn |f (x)|.
By Π 1 (R n ) we mean a set of polynomials in n variables of degree ≤ 1, i. e., of linear functions on R n . For x (0) ∈ R n , R > 0, by B(x (0) ; R) we denote the n-dimensional Let S −1 = (l ij ). Linear polynomials λ j (x) := l 1j x 1 + . . . + l nj x n + l n+1,j have a property λ j x (k) = δ k j . We call λ j the basic Lagrange polynomials corresponding to S. For an arbitrary x ∈ R n , These equalities mean that λ j (x) are the barycentric coordinates of x. For details, see [1, §1.1].
An interpolation projector P : C(Ω) → Π 1 (R n ) corresponds to a simplex S ⊂ Ω if the nodes of P coincide with the vertices of S. This projector is defined by the equalities P f x (j) = f x (j) . The following analogue of the Lagrange interpolation formula holds: Denote by P Ω the norm of P as an operator from C(Ω) in C(Ω). From (1), it follows that P Ω = max x∈Ω n+1 j=1 |λ j (x)|.
Since λ j (x) are the barycentric coordinates of a point x we have also Consider the case Ω = B := B(x (0) ; R). It is proved in [2] that If S is a regular simplex inscribed into the ball, then P B depends neither on the center x (0) nor on the radius R of the ball nor on the choice of such a simplex. In this case (see [2,Theorem 2]) Various geometric estimates concerning polynomial interpolation are given in [1]. In particular, this book contains the results corresponding to the linear interpolation on the unit cube Q n := [0, 1] n . Later some estimates for concrete n were improved (e. g., see [3], [4]). Interpolation by linear functions on a Euclidean ball in R n and related questions were considered in [5], [6], [2].
The points belong to the boundary sphere of B. The number of these points is equal to N = n+1 k , where k coincides with that number a or a + 1 on which ψ(t) takes a bigger value. Evidently, N is the number of (k − 1)-dimensional faces of an n-dimensional nondegenerate simplex. In Section 3 we discuss the questions related to the projector's norm invariance under an affine transform. In Section 4 we formulate a geometric conjecture the validity of which implies that the projector corresponding to a regular inscribed simplex has the minimal norm. This conjecture holds true at least for n = 1, 2, 3, 4.

The Maximum Points of λ(x) for a Regular Inscribed Simplex
By definition, put The detailed analysis and first values n and k are given in [2]. For these data and also the numbers N = n+1 k , see Table 1. For n = 1, 2, 3, we have k = 1. If n > 3, then √ n + 1 > 2, hence, Thus, for n > 3 holds k < n+1 2 . Since k is an integer, for all n ≥ 2 we have k ≤ n 2 . Suppose S is a regular simplex inscribed into a ball B, λ j are the basic Lagrange polynomials of this simplex, and P : C(B) → Π 1 (R n ) is the corresponding interpolation projector.
Proof. It is sufficient to consider some given ball B ⊂ R n , some regular simplex S inscribed into B, and also G = conv the equality (6) holds true. Note that in this trivial case a set of maximum points of the function λ(x) coincides with the ball B. Now let n ≥ 2. First note that the ceter of gravityc of the simplex S belongs to the segment [g, h]. Indeed, the equalities mean that (n + 1)c = kg + (n + 1 − k)h, пїЅ. пїЅ.
For proving (6), it is sufficient to indicate a linear polynomial p which takes in the nodes values ±1 and such that p(y) = P B . The equalities p x (j) = ±1 imply If p(y) = P B , then all the values in this chain coincide, therefore, λ(y) = P B .
Let us show that the above property is fulfilled for the polynomial p ∈ Π 1 (R n ) with values Since p is a linear function, from (7) and (9) it follows that p(g) = −1, p(h) = 1.
Making use of (8), we get The center of gravity of a regular inscribed simplex coincides with the center of the ball. The increment of p is proportional to the distance between the points, hence where R is the radius of the ball. Utilizing the found values, we obtain The value of the latter fraction does not depend on choice of a ball and a regular inscribed simplex. Let us calculate this value for the concrete S and B. Namely, as S we take the regular simplex with vertices The length of any edge of S is equal to In accordance to (7), the coordinates of g and h are From this, The simple calculation yeilds Thus, in this case Continuing (10), we can write If 1 ≤ k ≤ n+1 2 , then the last expression coincides with ψ(k). The noted inequality is true. Moreover, k coincides with that number a or a + 1 on which ψ(t) takes a bigger value. Therefore, p(y) = max{ψ(a), ψ(a + 1)} = P B .
The proof is complete. Theorem 2. In denotations of the previous theorem, [g, h] is the segment of maximal length in S parallel to vector gh.
Proof. In [7], the author obtained the calculation formulae for length and endpoints of the maximal segment in S of a given direction. One can apply these formulas for some simplex and take into account the similarity arguments. But it is much simpler to use the following characterization of the maximal segment proved in [7] (see there Lemmas 1 and 2). A segment in a simplex parallel to a given vector has maximal length iff every (n−1)-dimensional face of the simplex contains at least one endpoint of this segment.
Let the notation x = {β 1 , . . . , β n+1 } means that a point x has barycentric coordinates β 1 , . . . , β n+1 with respect to S. By G j denote (n − 1)-dimensional face of the simplex not containing the jth vertex. For points of G j , all barycentric coordinates are nonnegative and β j = 0. We have The number of nonzero barycentric coordinates in these equalities is equal to k and n+1−k respectively. Clearly, g ∈ G k+1 , . . . , G n+1 , h ∈ G 1 , . . . , G k . So, every (n−1)dimensional face of S contains an endpoint of the segment [g, h]. Consequently, this segment has maximal length of all the segments of given direction in S. Note that this argument is suitable for any simplex and k = 1, . . . , n.

The Projector's Norm Invariance under an Affine Transform
In 1948, F. John [8] proved that any convex body in R n contains a unique ellipsoid of maximum volume. Also he gave characterization of those convex bodies for which a maximal ellipsoid is the unit Euclidean ball B n (in details see, e. g., [9], [10]). John's theorem implies the analogous statement which characterizes a unique minimum volume ellipsoid containing a given convex body.
We shall consider a minimum volume ellipsoid containing a given nondegenerate simplex. For brevity, such an ellipsoid will be called it a minimal ellipsoid. Obviously, a minimal ellipsoid of a simplex is circumscribed around this simplex. The center of the ellipsoid coincides with the center of gravity of the simplex. A minimal ellipsoid of a simplex is a Euclidean ball iff this simplex is regular. This is equivalent to the well-known fact that the volume of a simplex contained in a ball is maximal iff this simplex is regular and inscribed into the ball (see, e. g., [11], [12], [13]).
By definition, put κ n := vol(B n ). Denote by σ n the volume of a regular simplex inscribed into the unit ball B n . Suppose S is an arbitrary n-dimensional simplex and E is the minimal ellipsoid of S. If a nondegenerate affine transform maps S into a regular simplex inscribed into B n , then the image of E under this transform coincides with B n . Hence, vol(E) vol(S) = κ n σ n .
It is known that (see, e. g., [14], [15], [1]). Therefore, vol(E) = K n vol(S), In addition, The value K n is included in the lower bound of the norm of a projector with nodes in B n . Let χ n (t) be the standardized Legendre polynomial of degree n: χ n (t) := 1 2 n n! (t 2 − 1) n (n) .
There exists a constant C > 0 not depending on n such that for any interpolation projector P : Inequalities (11) were obtained by the author in [6]. The right-hand estimate holds true, if we take, e. g., Assume S and S ′ are nondegenerate simplices in R n with vertices x (j) , . . . , x (n+1) and y (1) , . . . , y (n+1) respectively. Let S be the vertex matrix of S. Denote by Y the n × (n + 1)-matrix whose jth column contains the coordinates of y (j) . Let λ 1 , . . . , λ n+1 be the basic Lagrange polynomials of S. Lemma 1. There exists a unique affine transform F of space R n which maps S into S ′ and such that y (j) = F x (j) . The equality y = F (x) is equivalent to any relation    Proof. Each nondegenerate affine transform of R n has the form F (x) = A(x) + b, where A : R n → R n is a nondegenerate linear operator. Let A = (a ij ) be the matrix of the operator A in the canonical basis. In coordinate form, the equality y = A(x)+b is equivalent to the relation Define M as the n × (n + 1)-matrix standing on right-hand side. The conditions This means that an affine transform satisfying the conditions of the theorem is unique and has the form (12). Since λ j ∈ Π 1 (R n ) and λ j x (k) = δ k j , the equality (13) also gives an affine transform y = F (x) such that F x (k) = y (k) . From the uniqueness of F , it follows that (13) is equivalent to (12). This equivalence can be proved also directly. Let us rewrite (13) in the coordinate form using the coefficients of the polynomials λ j : Thus, (13) means that Since S −1 = (l ij ), these equalities are equivalent to (12). We note that the norm of an interpolation projector is invariant under an affine transform.

Theorem 3.
Suppose Ω is a convex body in R n containing a nondegenerate simplex S, Ω ′ and S ′ are their images under a nondegenerate affine transform, P : C(Ω) → Π 1 (R n ) and P ′ : C(Ω ′ ) → Π 1 (R n ) are interpolation projectors with the nodes in the vertices of S and S ′ respectively. Then P Ω = P ′ Ω ′ . Proof. Let x 1 , . . . , x n+1 be the vertices of the simplex S. We will assume that the verices of the simplex S ′ are numerated so that y (j) = F x (j) . Under this condition, the set of barycentric coordinates of an arbitrary point x ∈ R n with respect to S coincides with the set of barycentric coordinates of the point y = F (x) with respect to S ′ . This follows from the equalities The second equality coincides with relation (13) of Lemma 1. In accordance with formula (2), we have Corollary 1. Suppose S is a nondegenerate simplex with minimal ellipsoid E and S ′ is an arbitrary regular simplex inscribed into B n . If P : C(E) → Π 1 (R n ) and P ′ : C(B n ) → Π 1 (R n ) are the projectors having nodes in the vertices of S and S ′ respectively, then P E = P ′ Bn . Proof. Consider the nondegenerate affine transform which maps the simplex S into the regular simplex S ′ . This transform maps the ellipsoid E into the ball B n . It remains to apply Theorem 3 in the case when Ω is the minimal ellipsoid of S.
Let us supplement Corollary 1 with the following remark. Denote here by λ j the basic Lagrange polynomials of a simplex S. The maximum points of the function λ(x) = |λ j (x)| lying in the minimal ellipsoid E have the same geometric description that is formulated in Theorem 1. In the condition of this theorem, we must replace the regular simplex by an arbitrary one, and the circumscribed ball by the minimal ellipsoid of the simplex. At the specified points of the border of the ellipsoid, λ(x) takes maximal value equal to P E . This result can be established according to the scheme above.
Corollary 2. There exists a universal constant C > 0 such that for every ellipsoid E ⊂ R n and every interpolation projector having the nodes in E we have This follows immediately from (11) and Corollary 1.

On Some Extremal Property of a Regular Simplex Inscribed into a Ball
Consider a nondegenerate simplex S ⊂ R n . Let E be the minimal ellipsoid containing S. Fix a natural number m ≤ n 2 . To each set of m vertices of S assign the point y ∈ E defined as follows. Let g be the center of gravity of the (m − 1)-dimensional face of S containing the selected vertices, and let h be the center of gravity of the (n − m)-dimensional face containing the remaining n + 1 − m vertices. Then y is the intersection point of the straight line (gh) with the boundary of E in the direction from g to h.
Now we formulate the following conjecture.
(H1) For a given m ≤ n 2 and any nondegenerate simplex S ⊂ B n , there exists a set of m vertices of S such that y ∈ B n .
A stronger version of the hypothesis asserts that the specified property holds for any m ≤ n 2 (H2). For our purposes, it is sufficient that (H1) was true for m = k(n). The number k = k(n) is defined in Section 2.
Theorem 4. For m = 1 conjecture (H1) holds true. Proof. Suppose S is a simplex with vertices x (j) ∈ B n and the center of gravity c. The center of the minimal ellipsoid containing S also lies in c. Hence, in the case m = 1 the points y has the form y (j) = 2c − x (j) , j = 1, . . . , n + 1. We need to show that there exists a vertex x of the simplex such that 2c − x ≤ 1. Since S is nondegenerate, for some vertex x we have (c, x − c) ≥ 0. This means that i. e., the vertex x is suitable. The theorem is proved.
Denote by θ n (B n ) the minimal norm of an interpolation projector P : C(B n ) → Π 1 (R n ) with the nodes in B n . By P ′ denote a projector whose nodes coincide with the vertices of a regular simplex S ′ inscribed into B n .
Theorem 5. Suppose (H1) is true for m = k(n). Then θ n (B n ) = P ′ Bn . Proof. Consider an arbitrary projector P with the nodes x (j) ∈ B n . Let S be the simplex with these vertices and let λ j be the basic Lagrange polynomials corresponding to S. Denote by E the minimal ellipsoid of the simplex. Since S ⊂ B n , for some set consisting of k = k(n) vertices of the simplex the corresponding point y is contained in the ball. Let us fix y and write the following relations: |λ j (y)| ≤ max x∈Bn n+1 j=1 |λ j (x)| = P Bn .
We made use of the formula for the projector norm, Theorem 1, Corollary 1, and also the remark after this corollary. The inequality in the above chain follows from the condition y ∈ B n . Note that if y lies inside the ball, then this equality becomes strict. Therefore, for any projector with nodes in B n we have P ′ Bn ≤ P Bn . This implies that θ n (B n ) = P ′ Bn . The proof is complete. Corollary 3. If 1 ≤ n ≤ 4, then θ n (B n ) = P ′ Bn . Proof. In the case n = 1, the proposition is equivalent to the fact that the norm of an interpolation projector P : C[−1, 1] → Π 1 (R) becomes minimal for the projector having the nodes at the endpoints of the segment [−1, 1]. If 2 ≤ n ≤ 4, then k(n) = 1, and the required result follows immediately from Theorems 4 and 5.
Corollary 3 was proved in [2] by another method suitable only for dimensions n with the property k(n) = 1.
However, starting from n = 5, we have k(n) > 1 (see [2]). Nethertheless, the equality θ n (B n ) = P ′ Bn still can be obtained on the way directed by Theorem 5.
In the propositions of this section, the unit ball B n may be replaced by an arbitrary Euclidean ball B; this leads to the equivalent results.