MATHEMATICS

A. I. FET

STABILITY THEOREMS FOR CONVEX SURFACES CLOSE TO A SPHERE

(Presented by Academician S. L. Sobolev, 15 VI 1963)

1. To every uniqueness theorem in differential geometry there corresponds a certain stability theorem, asserting that if the conditions are satisfied with a prescribed error \(\varepsilon\), then the conclusion is also true with accuracy up to \(\varphi(\varepsilon)\). Only in a few cases is the function \(\varphi(\varepsilon)\) known or at least estimated. We wish to prove a stability theorem corresponding to Liebmann’s theorem on the rigidity of the sphere, generalized to the \(n\)-dimensional case. It is essential that, in this, the order of our estimate \(\varphi(\varepsilon)\) does not depend on the dimension \(n\).

In this note \(C_1, C_2, \ldots\) denote numbers depending only on \(n\).

1. We first prove a stability theorem corresponding to the known theorem ((\(^{1}\), p. 141; we were unable to consult the works cited in (\(^{1}\))): a convex body all of whose projections are balls (not necessarily of equal radius) is a ball.

Let \(T^n\) be a convex body in Euclidean space \(E^n\), and let \(R(T^n)\), \(r(T^n)\) be the radii of the smallest circumscribed and, respectively, largest inscribed ball of \(T^n\). Put
\[ \delta(T^n)=R(T^n)-r(T^n). \]
Denote by \(P^k(T^n)\) the orthogonal projection of \(T^n\) onto an arbitrary \(k\)-dimensional plane \(P^k \subset E^n\).

Theorem 1. If for all planes \(P^k\) of a given dimension \(k\) \((2 \le k \le n)\)
\[ \delta(P^k(T^n))<\varepsilon, \]
then
\[ \delta(T^n)<C\varepsilon, \]
where \(C\) is a constant depending only on \(n\).

Lemma 1. Let \(S_1^p, S_2^p\) be spheres of radii \(r_1, r_2\) in \(E^{p+1}\), \(0<C_1 \le r_1 \le r_2 \le C_2\), and let points \(a_1,b_1 \in S_1^p\), \(a_2,b_2 \in S_2^p\), with distance \(\rho(a_1,b_1)\ge C_3>0\), \(\rho(a_1,a_2)<\varepsilon\), \(\rho(b_1,b_2)<\varepsilon\). If \(S_1^p\) lies inside the sphere \(S_3^p\) of radius \(r_1+\varepsilon\), concentric with \(S_2^p\), then there exists \(C_4\) such that
\[ |r_1-r_2|<C_4\varepsilon . \]

Drawing a plane \(E^3 \subset E^{p+1}\) containing \(a_1,b_1\) and the centers of \(S_1^p,S_2^p\), one easily reduces the proof of the lemma to the case \(p=2\).

Proof of Theorem 1. It is enough to consider the case \(k=n-1\), since the general case is then proved by induction. Further, if \(\rho\) is the minimal value of \(R(P^{n-1}(T^n))\) over all \(P^{n-1}\), then \(T^n\) is contained in a cylinder of radius \(\rho\), and for any other \(P_1^{n-1}\) the width of the body \(P_1^{n-1}(T^n)\) is no greater than \(2\rho\). Therefore one may assume that every \(P^{n-1}\) contains a ball \(K^{n-1}(P^{n-1})\) of unit radius for which the Fréchet distance between the surfaces of the bodies is
\[ \rho\bigl(K^{n-1}(P^{n-1}),\,P^{n-1}(T^n)\bigr)<\varepsilon . \]

Construct the smallest ball \(K^n\) containing \(T^n\); by Jung’s known theorem, the radius of \(K^n\) does not exceed a certain \(C_2\). Further, there is a pair of contact points \(a,b\) of the body \(T^n\) with the boundary sphere \(S^{n-1}\) of the ball \(K^n\), the distance

between which is not less than \(\sqrt{2}/2\); indeed, otherwise by a suitable translation one could obtain from \(K^n\) a ball containing \(T^n\) strictly inside.

Construct some plane \(P_0^{n-1}\) containing \(a, b\) and the center of \(K^n\), and put \(P_0^{n-1}(K^n)=K^{n-1}\); obviously, the radii of \(K^n\) and \(K^{n-1}\) are equal, \(P_0^{n-1}(T^n)\subset K^{n-1}\), and the projections \(a', b'\) of the points \(a, b\) onto \(P_0^{n-1}\) are points of tangency of the boundaries of \(P_0^{n-1}(T^n)\), \(K^{n-1}\). Further,
\[ \rho\bigl(K^{n-1}(P_0^{n-1}), P_0^{n-1}(T^n)\bigr)<\varepsilon, \]
so that
\[ K^{n-1}(P_0^{n-1})\subset K_\varepsilon^{n-1}, \]
where \(K_\varepsilon^{n-1}\) is a ball, concentric with \(K^{n-1}\), of radius larger by \(\varepsilon\). On the other hand, \(a'\) and \(b'\) are at distance from the boundary of \(K^{n-1}(P_0^{n-1})\) not exceeding \(\varepsilon\) and belong to the boundary of \(K^{n-1}\); by Lemma 1, the radius of \(K^n\) is less than \(1+C_4\varepsilon\). Having constructed the largest ball \(K^{n'}\) contained in \(T^n\) and applying Lemma 1 again, we similarly find that the radius of \(K^{n'}\) is greater than \(1-C_5\varepsilon\), which proves the theorem.

1. Let us introduce a class of symmetrizations for convex surfaces in \(E^n\), containing, as particular cases, the known symmetrizations of Steiner and Schwarz. Let \(E^p\) be a plane in \(E^n\), and \(T^n\subset E^n\) a convex body. For any point \(x\in E^p\) construct the ball \(K_x^{n-p}\) with center at \(x\), lying in the orthogonal \((n-p)\)-dimensional plane \(E_x^{n-p}\) and having volume equal to the volume of \(T^n\cap E_x^{n-p}\). Put
   \[ S_{n,p}(T^n)=\bigcup_{x\in E^p}^{p} K_x^{n-p}; \]
   \(S_{n,1}\) is the Schwarz symmetrization, \(S_{n,n-1}=S\) is the Steiner symmetrization.

It is easy to see that every symmetrization \(S_{n,p}\) reduces to a finite number of Steiner symmetrizations with respect to suitable \((n-1)\)-dimensional planes, followed by a passage to the limit. Hence, in particular, it follows that \(S_{n,p}\) preserves volume, does not increase diameter, and that Theorem 4 \((^2)\) remains valid when \(S\) is replaced by \(S_{n,p}\).

1. The following theorem can be used to prove a number of stability theorems in which the surfaces turn out to differ little from a sphere.

Theorem 2. Let \(\Phi\) be a class of convex surfaces in \(E^n\) possessing the following properties:

1) The diameter of a surface of the class \(\Phi\) is not greater than \(2+\varepsilon\), and its width is less than \(2-\varepsilon\).

2) The class \(\Phi\), together with every surface \(F^{n-1}\), contains the surface \(S(F^{n-1})\) obtained by Steiner symmetrization.

3) The class \(\Phi\), together with a sequence of surfaces \(F_i^{n-1}\), contains their limit surface.

Then every surface of the class \(\Phi\) is contained in a ball of radius \(1+C\sqrt{\varepsilon}\) and contains a ball of radius \(1-C\sqrt{\varepsilon}\), where \(C\) depends only on \(n\).

Proof. First we prove

Lemma 2. The orthogonal projection of a surface \(F^{n-1}\) onto an arbitrary two-dimensional plane \(P^2\) has area not less than \(\Pi-\varepsilon\). Let the area of \(P^2(F^{n-1})\) be equal to \(\sigma\) and \(\Pi\tau^2=\sigma\).

Construct the plane \(P^{n-2}\), orthogonal to \(P^2\), and perform the symmetrization \(S_{n,n-2}\) with respect to \(P^{n-2}\); according to item 3,
\[ F_1^{n-1}=S_{n,n-2}(F^{n-1})\in\Phi. \]
It is clear that all sections of \(F_1^{n-1}\) parallel to \(P^2\) have area not greater than \(\sigma\); therefore the radii of all \(K_x^{n-2}\) are not greater than \(\tau\). Thus, the distances of all points of \(F_1^{n-1}\) from \(P^{n-2}\) do not exceed \(\tau\). If now one constructs some plane \(P^{n-1}\supset P^{n-2}\) and two parallel \(P^{n-1}\)-planes \(P_1^{n-1}, P_2^{n-1}\) on different sides and at distance \(\tau\) from \(P^{n-1}\), then \(F_1^{n-1}\) lies between \(P_1^{n-1}, P_2^{n-1}\). But then the width
\[ \Delta(F_1^{n-1})\le 2\tau; \]
on the other hand, \(F_1^{n-1}\in\Phi\), consequently,

\(\Delta(F_1^{n-1}) \geqslant 2-\varepsilon\), so that \(\tau \geqslant 1-\varepsilon/2\), whence the assertion of the lemma follows.

To prove Theorem 2, let us note that for any \(F^{n-1} \in \Phi\) and any \(P^2\) the width of \(P^2(F^{n-1})\) in any direction does not exceed \(2+\varepsilon\); hence, as is known, it follows that the length of the perimeter of the domain \(P^2(F^{n-1})\) satisfies \(L \leqslant \pi(2+\varepsilon)\). By Lemma 2, the area of \(P^2(F^{n-1})\) is \(F \geqslant \pi-\varepsilon\). From Bonnesen’s inequality ((1), p. 83) it follows that

\[ 4\pi(P-\rho)^2 \leqslant L^2 - 4\pi F, \]

where \(P, \rho\) are the radii of concentric circles respectively containing \(P^2(F^{n-1})\) and contained in \(P^2(F^{n-1})\). Thus \(P-\rho < C_6\sqrt{\varepsilon}\); from Lemma 2 we find that \(P \geqslant 1-\varepsilon/2\), and it remains to apply Theorem 1. Let us note that, for small \(\varepsilon\), one may take for \(C\) any number greater than \(\sqrt{8\pi}\).

1. From Theorem 2, in particular, one can derive

Theorem 3. Let the Gaussian curvature of a convex surface \(F^{n-1}\) be between \(1-\varepsilon\) and \(1+\varepsilon\). Then \(F^{n-1}\) is contained in a ball of radius \(1+C\sqrt{\varepsilon}\) and contains a ball of radius \(1-C\sqrt{\varepsilon}\), where \(C\) depends only on \(n\).

Proof. Consider the class \(\Phi_1\) of convex surfaces satisfying the following conditions:

1. The diameter of the surfaces \(\Phi_1\) is not greater than \(2+C_7\varepsilon\).
2. The volume of the surfaces \(\Phi_1\) is not less than \(\chi_n-C_7\varepsilon\), where \(\chi_n\) is the volume of the unit \(n\)-dimensional ball.
3. The Gaussian curvature of the surfaces \(\Phi_1\) is not less than \(1-C_7\varepsilon\).

To prove Theorem 3 it suffices to verify two assertions:

a) the class \(\Phi_1\) satisfies the conditions of Theorem 2 with some \(C_8\varepsilon\) in place of \(\varepsilon\);

b) every surface satisfying the conditions of Theorem 3 belongs to the class \(\Phi_1\) for some \(C_7\).

The proof of a) and b) is based on results of V. I. Diskant \((^3)\). From Theorem 3 and the remark in item (4) \((^3)\) it follows that the width \(\Delta(F^{n-1}) \geqslant 2-C_9\varepsilon\) for all \(F^{n-1}\in\Phi_1\), so that condition 1) of our Theorem 2 is satisfied with \(\max(C_7,C_9)\) in place of \(\varepsilon\). From Theorem 4 \((^2)\) it follows that conditions 2), 3) are also satisfied, and (a) is proved. b) follows from Theorem 2 \((^3)\) and Theorem 3 \((^2)\).

I consider it my duty to express my gratitude to V. A. Toponogov, who noted that the theorems stated in \((^2)\) may be useful in the question of stability.

Institute of Mathematics with Computing Center
Siberian Branch of the Academy of Sciences of the USSR

Received
7 VI 1963

REFERENCES

1. T. Bonnesen, W. Fenchel, Theorie der konvexen Körper, Berlin, 1934.
2. A. I. Fet, DAN, 153, No. 2 (1963).
3. V. I. Diskant, DAN, 153, No. 3 (1963).