V. I. DISKANT

STABILITY IN LIEBMANN’S THEOREM

(Presented by Academician S. L. Sobolev, 4 V 1964)

1. A stability theorem corresponding to Liebmann’s theorem on the rigidity of the sphere was proved by A. I. Fet in (⁴). In the present work an exact estimate is given for the order of the stability function \(\varphi(\varepsilon)\) (see (⁴)) with respect to \(\varepsilon\). Namely, we prove

Theorem 1. If the Gaussian curvature \(K\) of a convex \(n\)-dimensional surface \(\Gamma\) satisfies the inequalities \(1-\varepsilon \leqslant K \leqslant 1+\varepsilon\), then for the radius \(r\) of the largest inscribed ball and for the radius \(R\) of the smallest circumscribed ball the estimates

\[ r \geqslant 1-C\varepsilon,\qquad R \leqslant 1+C\varepsilon, \]

hold, where \(C\) is a number depending only on \(n\), \(0 \leqslant \varepsilon \leqslant \tfrac12\).

The sharpness of the estimate follows from consideration of spheres of curvature \(1-\varepsilon\) and \(1+\varepsilon\).

2. We first note two lemmas for a surface of revolution.

Lemma 1. If the Gaussian curvature \(K\) of a convex surface \(\Sigma\), formed by the rotation of a curve \(y=y(x)\) \((y \geqslant 0)\) about the \(x\)-axis, satisfies the inequality \(K \geqslant a^n\), where \(a>0\), then \(\max y(x) \leqslant 1/a\).

Proof. Let \(\max y(x)=y(0)\). Suppose that \(y(0)>1/a\). Consider the curve

\[ x=x_a(y)= \begin{cases} \displaystyle \int_y^{y(0)} \frac{\bigl[a^n\bigl(y^n-y^n(0)\bigr)+1\bigr]^{1/n}} {\sqrt{1-\bigl[a^n\bigl(y^n-y^n(0)\bigr)+1\bigr]^{2/n}}}\,dy, & \text{if } y\ne y(0),\\[2ex] 0, & \text{if } y=y(0). \end{cases} \]

The surface of revolution of the curve \(x=x_a(y)\) about the \(x\)-axis has curvature \(K(x)\equiv a^n\). Applying Lemma 1 (³) to the curves \(x=x_a(y)\) and \(x=x(y)\), we obtain \(x'(\bar y)=0\), where \(\bar y=\sqrt[n]{\,y^n(0)-1/a^n\,}>0\). The latter is impossible.

Lemma 2. If the curve \(y=y(x)\) is defined on \([0,\eta]\) and satisfies the conditions

\[ y(x)\geqslant 0,\qquad y'(x)\leqslant 0,\qquad y''(x)<0,\qquad y'(\eta)=0,\qquad y(0)\leqslant 1/a, \]
\[ K(x)\geqslant a^n, \]

then all points of this curve belong to the figure \(G\), whose boundary consists of the curve \(x=x_a(y)\) \((0\leqslant y\leqslant y(0))\), the segments \([0,x_a(0)]\) of the \(x\)-axis, and \([0,y(0)]\) of the \(y\)-axis.

Lemma 2 is a direct consequence of Lemma 1 (³).

3. The assertion of Theorem 1 concerning the radius \(r\) follows from the following estimate.

Proposition A. If the Gaussian curvature \(K\) of a convex surface \(\Gamma\) satisfies the inequalities \(a^n \leqslant K \leqslant b^n\), then for the radius \(r\) of the largest inscribed ball the estimate

\[ r \geqslant A+\bigl(B^{n+1}-A^{n+1}\bigr)\, \frac{(n-1)\chi_{n+1}}{\chi_n} \left[ \frac{1}{A^n}+\frac{1}{B^n}\left(\frac{A}{B}\right)^{n^2+n+1} \right], \tag{1} \]

holds, where \(\chi_n\) is the volume of the \(n\)-dimensional unit ball, \(A=1/a\), \(B=1/b\).

The proof of Proposition A is based on the following lemma.

Lemma 3. Let the convex body \(T\) be bounded by a surface of revolution \(\Sigma\) about the \(x\)-axis. Suppose that the Gaussian curvature \(K\) of the surface \(\Sigma\) satisfies the inequality \(K \geq a^n\) and that the volume of the body \(V_T \geq \varkappa_{n+1}B^{n+1}\). Then, for the distance \(\rho(Z,X_0)\) between the center of gravity \(Z=(z,0,\ldots,0)\) of the body \(T\) and the point of intersection \(X_0=(x(0),0,\ldots,0)\) of the \(x\)-axis with the surface \(\Sigma\), the estimate
\[ \rho(Z,X_0)\geq A+(B^{n+1}-A^{n+1})\frac{\varkappa_{n+1}(n+1)}{\varkappa_n} \left[\frac{1}{A^n}+\frac{1}{B^n}\left(\frac{A}{B}\right)^{n^2+n+1}\right]. \tag{2} \]

Proof of Lemma 3. Let \(\Sigma\) be obtained by rotating the curve \(y=y(x)\) (\(y\geq 0\)). Suppose that \(y(0)=\max y(x)\). A suitable approximation allows us to assume that \(y=y(x)\) satisfies all the conditions of Lemma 1 and, for \(x\geq 0\), the conditions of Lemma 2, for, by Lemma 1, \(y(0)\leq A\). Consider in the \(xy\)-plane the figure \(M\), composed of the figure \(G\) and the figure symmetric to \(G\) with respect to the \(y\)-axis.

Introduce the following notation: \(T_M\) is the body obtained by rotating the figure \(M\) about the \(x\)-axis; \(V_M\) is the volume of \(T_M\); \(T_{\mathrm{r}}\) (\(T_{\mathrm{l}}\)) is the part of the body \(T\) lying in the half-space \(x\geq 0\) (\(x\leq 0\)); \(V_{\mathrm{r}}\) (\(V_{\mathrm{l}}\)) is the volume of \(T_{\mathrm{r}}\) (\(T_{\mathrm{l}}\)); \(T_{\mathrm{r}}^M\) (\(T_{\mathrm{l}}^M\)) is the part of the body \(T_M\) lying in the half-space \(x\geq 0\) (\(x\leq 0\)); \(T_{\mathrm{r}}^0=T_{\mathrm{r}}^M\setminus T_{\mathrm{r}}\), \(T_{\mathrm{l}}^0=T_{\mathrm{l}}^M\setminus T_{\mathrm{l}}\); \(V_{\mathrm{r}}^0\) (\(V_{\mathrm{l}}^0\)) is the volume of \(T_{\mathrm{r}}^0\) (\(T_{\mathrm{l}}^0\)); \(z_{\mathrm{r}}\) (\(z_{\mathrm{l}}\)) is the abscissa of the center of gravity of \(T_{\mathrm{r}}\) (\(T_{\mathrm{l}}\)); \(z_{\mathrm{r}}^M\) (\(z_{\mathrm{l}}^M\)) is the abscissa of the center of gravity of \(T_{\mathrm{r}}^M\) (\(T_{\mathrm{l}}^M\)); \(z_{\mathrm{r}}^0\) (\(z_{\mathrm{l}}^0\)) is the abscissa of the center of gravity of \(T_{\mathrm{r}}^0\) (\(T_{\mathrm{l}}^0\)).

Since \(\rho(Z,X_0)=x(0)-z=|x(0)-z|\geq x(0)-|z|\), it suffices to find the required estimates for \(x(0)\) and \(|z|\).

Let us estimate \(|z|\). By the additivity property of the center of gravity we have
\[ \frac{V_M}{2}z_{\mathrm{r}}^M=V_{\mathrm{r}}z_{\mathrm{r}}+V_{\mathrm{r}}^0z_{\mathrm{r}}^0 \quad \left(\frac{V_M}{2}z_{\mathrm{l}}^M=V_{\mathrm{l}}z_{\mathrm{l}}+V_{\mathrm{l}}^0z_{\mathrm{l}}^0\right); \tag{3} \]
\[ V_Tz=V_{\mathrm{l}}z_{\mathrm{l}}+V_{\mathrm{r}}z_{\mathrm{r}}. \tag{4} \]
From (3) and (4) we find
\[ z=\frac{-V_{\mathrm{l}}^0z_{\mathrm{l}}^0-V_{\mathrm{r}}^0z_{\mathrm{r}}^0}{V_T}; \tag{5} \]
from the condition of the lemma
\[ V_T\geq \varkappa_{n+1}B^{n+1}. \tag{6} \]
Applying Theorem 2 (?) to the sphere of radius \(A\) and to the surface of the body \(T_M\), we obtain
\[ V_T\leq V_M\leq \varkappa_{n+1}A^{n+1}. \tag{7} \]
From (6) and (7) we have
\[ V_{\mathrm{r}}\leq \frac{V_M}{2}\leq \frac12\varkappa_{n+1}A^{n+1} \quad \left(V_{\mathrm{l}}\leq \frac12\varkappa_{n+1}A^{n+1}\right); \]
therefore
\[ V_{\mathrm{r}}=V_T-V_{\mathrm{l}}\geq \varkappa_{n+1}B^{n+1}-\frac12\varkappa_{n+1}A^{n+1} \quad \left(V_{\mathrm{l}}\geq \varkappa_{n+1}B^{n+1}-\frac12\varkappa_{n+1}A^{n+1}\right), \]
whence
\[ V_{\mathrm{r}}^0\leq \varkappa_{n+1}(A^{n+1}-B^{n+1}) \quad \left(V_{\mathrm{l}}^0\leq \varkappa_{n+1}(A^{n+1}-B^{n+1})\right). \tag{8} \]

To estimate \(z_{\mathrm{r}}^0\) (\(z_{\mathrm{l}}^0\)), note that \(z_{\mathrm{r}}^0\leq x_a(0)\). For \(x_a(0)\) the inequality
\[ \frac{1}{n+1}\varkappa_n y^n(0)x_a(0)\leq \frac{\varkappa_{n+1}A^{n+1}}{2}, \tag{9} \]
is valid, in whose left-hand side stands the volume of the cone contained in \(T_{\mathrm{r}}^M\). Obviously, \(y(0)\geq \Delta/2\), where \(\Delta\) is the width of the body \(T\).

Replace in Theorem 3 \((^3)\) the estimate for \(V_S\) in formula (7) by the inequality
\[ V_S \ll 2V_C \ll \varkappa_{n+1} A^n x(0). \tag{10} \]
Then in Theorem 3 \((^3)\) for \(\Delta\) we obtain the estimate
\[ \Delta \gg 2B\left(\frac{B}{A}\right)^n . \tag{11} \]

Let us prove (10). Note that \(V_C\) is the volume of a segment of a ball of radius \(A\), \(x(0)\) \((x(0)\ll A)\) is the height of the segment. Denote by \(S\) the area of the base of the segment. Put \(V_C=hP\) and \(\frac12\varkappa_{n+1}A^{n+1}-V_C=h'Q\), where \(h\) is the height of the segment and \(h'=A-h\). Then \(P\ll S\ll Q\), whence (10) follows.

Using (10), we find from (9)
\[ z_{\Pi}^{0}\ll \frac{(n+1)\varkappa_{n+1}}{2\varkappa_n}\, B\left(\frac{A}{B}\right)^{n^2+n+1}. \tag{12} \]

By the method which we used in the proof of Theorem 3 \((^3)\), one can show that
\[ x(0)\gg A+ \frac{\varkappa_{n+1}(n+1)}{\varkappa_n A^n} \left(B^{n+1}-A^{n+1}\right). \tag{13} \]

From inequalities (6), (8), (12), (13) the estimate (2) follows.

Proof of Proposition A. Denote by \(\Pi\) the center of gravity of the body bounded by \(\Gamma\). Let \(m=\min_L\rho(\Pi,L)\), and let \(L_m\) be a supporting plane for which \(\rho(\Pi,L_m)=m\). Suppose that the \(x\)-axis is the outer normal to \(L_m\). With the aid of Steiner symmetrizations and a limiting passage one can obtain from \(\Gamma\) a surface of revolution \(\Sigma\) about the \(x\)-axis. For the surface \(\Sigma\) the conditions of Lemma 3 are fulfilled. Indeed, \(V_\Sigma=V_\Gamma\), and by Theorem 3 \((^2)\) \(V_\Gamma\gg\varkappa_{n+1}B^{n+1}\). Moreover, by Theorem 4 \((^2)\) the Gaussian curvature of \(\Sigma\) is not less than \(a^n\). Denote by \(Z=(z,0,\ldots,0)\) the center of gravity of the body bounded by \(\Sigma\), and by \(L_\Sigma\) the supporting plane to \(\Sigma\) for which the \(x\)-axis is the outer normal. Then \(\rho(Z,L_\Sigma)=x(0)-z\). On the other hand, one can show that \(\rho(Z,L_\Sigma)=\rho(\Pi,L_m)\). Consequently, \(m=x(0)-z\). But \(r\gg m_1\), since otherwise a ball of radius \(\bar r>r\) could be inscribed in \(T\) \((^1,\) p. 24). Thus,
\[ r\gg x(0)-z. \]

1. The assertion of Theorem 1 concerning the radius \(R\) follows from the following estimate.

Proposition B. If the Gaussian curvature \(K\) of the convex surface \(\Gamma\) satisfies the inequalities \(a^n\ll K\ll b^n\), then for the radius \(R\) of the minimal circumscribed ball the estimate
\[ R \ll 2B-A+\frac{B\varkappa_{n+1}}{\varkappa_n} \left[\left(\frac{A}{B}\right)^{(n+1)^2}-1\right]+ \]
\[ +\left(A^{n+1}-B^{n+1}\right) \frac{(\chi+1)\varkappa_{n+1}}{\varkappa_n} \left[ \frac{1}{A^n}+\frac{1}{B^n} \left(\frac{A}{B}\right)^{n^2+n+1} \right]. \]

Proof. If \(\bar R\) is the radius of the minimal circumscribed ball whose center coincides with the center of the maximal inscribed ball, then \(D\gg r+\bar R\) and \(R\ll\bar R\). Using estimate (1) for \(r\) and the estimate for \(D\) in Theorem 2 \((^3)\), we obtain the estimate in Proposition B.

The author expresses gratitude to A. I. Fet for posing the problem and for valuable guidance in the course of work on it.

Received
9 IX 1963

CITED LITERATURE

\(^{1}\) 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).
\(^{4}\) A. I. Fet, DAN, 153, No. 3 (1963).