UDC 513

MATHEMATICS

Academician A. D. ALEKSANDROV

ON THE MEAN VALUES OF THE SUPPORT FUNCTION

1. Let \(G\) be a convex body in \(n\)-dimensional Euclidean space; \(h(x,\nu)\) the distance from the point \(x \in G\) to the supporting plane to \(G\) with outward normal \(\nu\); \(\Omega\) the unit sphere (the set of all \(\nu\)); \(\chi_n\) its area. Define the mean values of the function \(h(x,\nu)\)

\[ h_k(x)=\left[\frac{1}{\chi_n}\int_{\Omega} h^{-k}(x,\nu)\,d\nu\right]^{-1/k},\qquad k\ne 0, \tag{1} \]

\[ h_0(x)=\exp \frac{1}{\chi_n}\int_{\Omega}\ln h(x,\nu)\,d\nu. \tag{2} \]

These functions \(h_k\) deserve attention not only because interesting geometric problems are connected with them, but also because they majorize, up to a factor, the solutions \(u(x)\) of second-order equations of a fairly general form under \(u|_{\partial G}=0\) \((^{1,2})\). In \((^{2})\), for linear equations the majorants are given in the form \(Hh_k(x)\) with one or another \(k\), \(0\le k\le n\), where \(H\) is expressed in terms of certain norms of the coefficients (and of the solution itself, if uniqueness conditions are not satisfied). Further, in \((^{3})\) it is proved that the majorants with \(h_0\) and \(h_n\) are sharp. Thus, for example, the function \(h_0\) admits a definition through solutions of elliptic linear second-order equations: \(h_0(s)=\sup |u(x)|\), where the supremum is taken over the solutions (with \(u|_{\partial G}=0\)) of all equations for which \(H=1\).

1. Let us list some of the simplest properties of the function \(h_k\).

(2.1) If \(G\subset G'\), \(G\ne G'\), then for every \(k\) and every \(x\in G\)
\[ h_k(x)<h'_k(x). \]

(2.2) If \(k>l\), then \(h_k(x)<h_l(x)\), except in the case when \(G\) is a ball and \(x\) is its center. (This follows from the well-known property of means; see, for example, \((^{7})\).)

(2.3) If \(k>-1\), then everywhere inside \(G\), \(d^2h_k<0\), while for \(k<-1\), \(d^2h_k>0\). For the proof we note that
\[ h(x,\nu)=h(x_0,\nu)+(x_0-x)\nu. \]
Substituting this in (1) or (2), it is easy to find \(d^2h_k\) and to verify the asserted statement. In addition, we find that \(h_{-1}(x)=\mathrm{const}\).

(2.4) As is known, if from the point \(x\) in each direction \(\nu\) one lays off a segment of length \(r(x,\nu)=h^{-1}(x,\nu)\), then one obtains the convex body \(G^x\), polar to \(G\) with respect to the point \(x\) (with respect to the unit ball with center \(x\)). Therefore, for \(k\ne 0\),

\[ h_k(x)=\left[\frac{1}{\chi_n}\int_{\Omega} r^k(x,\nu)\,d\nu\right]^{-1/k}. \tag{3} \]

Hence it is easy to conclude that if \(k\) is an integer, \(1\le k\le n\), then

\[ h_k(x)=\tau_k^{1/k}\hat V_k^{-1/k}(x), \tag{4} \]

where \(\tau_k\) is the volume of the \(k\)-dimensional unit ball, and \(\hat V_k(x)\) is the mean value of the \(k\)-dimensional volumes of the sections of the body \(G^x\) by \(k\)-dimensional planes passing through \(x\); in particular, \(\hat V_n(x)\) is the volume of \(G^x\). (The mean is understood here and below as the arithmetic mean in the sense of the natural measure in the set of ...

set of planes of the given dimension passing through a fixed point.)

What has been said makes it possible, among other things, to find \(h_n(x)\) when the polar body \(G^x\) is simply determined. For example, the body polar to an ellipsoid is an ellipsoid. Its volume is found from elementary geometric considerations. Thus we find that for an ellipsoid
\[ h_n(x)=(a_1,\ldots,a_n)^{1/n}(1-\rho^2(x))^{(n+1)/2n}, \]
where \(a_i\) are the semiaxes, and \(\rho(x)\) is the ratio of the distance from the center to \(x\) to the radius in the same direction.

1. Let us indicate estimates for \(H_k\ne \max h_k\) in terms of the following quantities: \(V_m\)—the greatest of the volumes of the \(m\)-dimensional sections of the body \(G\); \(W_m\)—the mean value of the volumes of its \(m\)-dimensional projections; \(w_m\)—the least of these volumes. In particular, \(V_n=W_n=w_n\) is the volume of \(G\), \(V_1\) is the diameter, \(W_1\) is the mean width. (Some inequalities between these quantities and the relation of \(W_m\) to the integral curvatures can be found in \((^4,^5)\). In particular
   \[ W_{n-1}=\frac{\tau_{n-1}}{\varkappa_n}S, \]
   where \(S\) is the area of \(\partial G\), \(\tau_m^{-1/m}W_m^{1/m}\ge \tau_{m+1}^{-1/(m+1)}W_{m+1}^{1/(m+1)}\).) We shall denote by \(\alpha_m\), etc., positive numbers depending only on \(m\) and the dimension \(n\).

(3.1) For every \(k\), obviously, \(H_k<V_1\). On the other hand, for \(k<1\), \(H_k>\alpha_kV_1\), but for \(k\ge 1\) such a lower estimate is impossible. (It suffices to note that \(G\) contains a segment of length \(V_1\), and therefore, if \(H_k'\) is \(\max h_k\) for such a segment, then \(H_k>H_k'\). Computing \(H_k'\), we obtain what was asserted.)

(3.2) For \(k>-1\), \(H_k<\frac12 W_1\), except in the case when \(G\) is a ball. By definition,
\[ W_1=\frac1{\varkappa_n}\int\bigl(h(x,\nu)+h(x,-\nu)\bigr)\,d\nu =\frac2{\varkappa_n}\int h(x,\nu)\,d\nu=2h_{-1}(x). \]
Therefore \(h_{-1}(x)=\frac12 W_1\), and the assertion follows from (2.2). On the other hand, it also follows from (2.2) that for \(k<-1\), \(h_k(x)>h_{-1}(x)\), again except in the case of a ball. Therefore, excluding this case, for \(k<-1\)
\[ \min h_k(x)>\frac12 W_1 \]
(cf. this with (2.3)).

(3.3)
\[ \alpha_n V_n^{1/n}\ge H_n\ge \beta_n V_n^{1/n}. \]

By virtue of (4) \(H_n=\tau_n^{1/n}\widetilde V_n^{-1/n}\), where \(\widetilde V_n=\min_x \widetilde V_n(x)\) is the least of the volumes of the polar bodies \(G^x\). Therefore (3.3) is equivalent to
\[ \lambda_n\le V_n\widetilde V_n\le \mu_n,\qquad \lambda_n=\tau_n\alpha_n^{-n},\quad \mu_n=\tau_n\beta_n^{-n}. \tag{5} \]
The fact that such inequalities for \(V_n\widetilde V_n\) hold is known and is proved simply. From the known properties of mutually polar bodies there follows the affine invariance of the product \(V_n\widetilde V_n\). Therefore it is enough to consider bodies of volume \(V_n=1\) contained in a given cube. Then it is obvious that the product \(V_n\widetilde V_n\) attains finite and positive maximum and minimum values, and this is precisely (5). It is not difficult also to obtain some values for \(\lambda_n,\mu_n\), but the question of the best values remains open (see, for example, \((^6)\), where \(\lambda_n=4^n(n!)^{-2}\) is given). It is probable that the greatest value of \(V_n\widetilde V_n\) is attained for an ellipsoid, and the least for a simplex, to which, accordingly, the best values of \(\mu_n,\lambda_n\) correspond.

(3.4) If \(1\le k<n\) and \(l\) is the integer part of \(k\), then
\[ \alpha_k' W_l^{1/l}\ge H_k\ge \beta_k' W_{l+1}^{1/(l+1)}, \]
\[ \alpha_k'' V_l^{1/l}\ge H_k\ge \beta_k'' V_{l+1}^{1/(l+1)} \quad\text{and}\quad H_k\ge \gamma_k w_l, \]
but lower estimates in terms of \(W_l\), \(V_l\) are impossible also for \(k=l\).

Let us prove the first inequality: \(H_k\le \alpha_k W_l^{1/l}\). Since for \(k\ge l\), \(H_k\le H_l\), one may put \(k=l\). Let \(E\) be an \(k\)-dimensional plane, pro-

passing through the center of the sphere \(\Omega\); \(x_E\) is the projection of the point \(x\) onto \(E\); \(G_E\) is the projection of \(G\). For \(v\in \Omega\cap E\), obviously, \(h(x,v)=h(x_E,v)\), that is, the distance from \(x_E\) to the supporting plane of \(G_E\) with normal \(v\). Therefore, if we put

\[ h_k(x,E)=\left[\frac{1}{\chi_k}\int_{\Omega\cap E} h^{-k}(x,v)\,dv\right]^{-1/k}, \tag{6} \]

then \(h_k(x,E)\) will be nothing other than the function \(h_k\) for \(G_E\). Hence, according to (3.3), we conclude that \(h_k(x,E)=h_k(x_E,E)\leq \alpha_k V_k^{1/k}(E)\), where \(V_k(E)\) is the volume of \(G_E\).

If the measure in the set \(\mathscr E\) of planes \(E\) is normalized so that \(\operatorname{mes}\mathscr E=1\), then for every integrable \(f(v)\) we have

\[ \frac{1}{\chi_n}\int_{\Omega} f(v)\,dv = \int_{\mathscr E}\left(\frac{1}{\chi_k}\int_{\Omega\cap E} f(v)\,dv\right)dE. \]

Applying this equality to \(f(v)=h^{-k}(x,v)\) and using (6), we obtain

\[ h_k^{-k}(x) = \frac{1}{\chi_n}\int_{\Omega} h^{-k}(x,v)\,dv = \int_{\mathscr E}\left(\frac{1}{\chi_k}\int_{\Omega\cap E} h^{-k}(x,v)\,dv\right)dE = \int_{\mathscr E} h_k^{-k}(x,E)\,dE, \]

and since \(h_k(x,E)\leq \alpha_k V_k^{1/k}\), i.e. \(h_k^{-k}(x,E)\geq \alpha_k^{-k}V_k^{-1}(E)\), it follows that

\[ h_k^{-k}(x)\geq \alpha_k^{-k}\int_{\mathscr E}V_k^{-1}(E)\,dE \geq \alpha_k^{-k}\left(\int_{\mathscr E}V_k(E)\,dE\right)^{-1} = \alpha_k^{-k}W_k^{-1}, \]

where we have used the well-known inequality for means and the definition of the quantity \(W_k\). Thus, for every \(x\in G\) we have \(h_k(x)\leq \alpha_k W_k^{1/k}\), as was required to prove.

We omit the proofs of the other inequalities (3.4). The inequalities with \(W\) and \(V\) follow from one another, since \(\delta_m\geq W_m V_m^{-1}\geq \delta_m'\); moreover, \(W_{m+1}\geq \delta_m'' w_m\).

1. We now indicate functions estimating \(h_k(x)\), \(0<k\leq n\).

(4.1) Let \(G_m\) be the body symmetric to \(G\) with respect to the point \(x\in G\), and let \(V(x)\) be the volume of the body \(H=G\cap G_x\). It turns out that for \(h_n(x)\)

\[ \bar{\alpha}_n V_n^{1/n}(x)\geq h_n(x)\geq \beta_n V^{1/n}(x), \tag{7} \]

where, incidentally, \(\beta_n\) is the same as in (3.3). (From the left inequality (7) it is not difficult to conclude that if \(\bar V(x)\) is the smallest of the volumes cut off from \(G\) by arbitrary planes passing through \(x\), then \(h_n(x)\leq \alpha_n 2^{1/n}\bar V^{1/n}(x)\).)

Let us prove (7). Let \(\bar h_n\) be the function \(h_n\) for the body \(H\). Obviously, \(h_n(x)\geq \bar h_n(x)\), and by (3.3) \(\bar h_n(x)\geq \beta_n V^{1/n}(x)\), whence the right-hand inequality (7) follows.

For the proof of the left-hand inequality we use the lemma:

Lemma. If \(K\) is a convex body, \(K_x\) is the body symmetric to it with respect to the point \(x\in K\), and \(\bar K\) is the convex hull of \(K\cup K_x\), then

\[ V(K)\geq \gamma_n V(\bar K). \tag{8} \]

For the proof, observe that, by the affine invariance of the ratio of volumes, one may assume \(V(K)=1\) and that \(K\) is contained in some ball. On the set of bodies \(K\) satisfying these conditions, for all \(x\in K\), the volumes \(V(\bar K)\) are bounded. If \(\gamma_n^{-1}\) is their least upper bound, then (8) holds.

We apply this lemma to \(K=\widetilde G\), where \(\widetilde G\) is the body polar to \(G\) with respect to the point \(x\). Then \(K_x=\widetilde G_x\) is the body polar to \(G_x\). From the properties of polarity we conclude that \(\bar K\) is the body \(\widetilde H\), polar to \(H=G\cap G_x\) (because under polarity the common points of bodies correspond to planes not intersecting their polars). Therefore (8) gives:

\[ V(\widetilde G)\geq \gamma_n V(\widetilde H). \tag{9} \]

But, as a consequence of (4), \(\bar V(\bar G)=\tau_n\bar h_n^{-n}(x)\), \(\bar V(\bar H)=\tau_n\bar h_m^{-n}(x)\). Therefore, from (9) it follows that \(\bar h_n(x)\geq \gamma_n^{1/n}h_n(x)\), and since, by (3.3), \(\bar h_n(x)\leq \bar a_n V^{1/n}(x)\), we obtain the left-hand inequality (7). (The best values of the constants in (7), (8) are unknown. The best \(\gamma_n\) is the minimum of \(V(\bar K):V(K)\); it is probably attained when \(K\) is a simplex and \(x\) is one of its vertices.)

(4.2) Let \(h(x)\) be the distance from \(x\in G\) to \(\partial G\), i.e.
\[ h(x)=\min_v h(x,v). \]

For every \(k>0\) and \(\leq n\),
\[ h_k(x)\leq h^{1-n/k}(x)h_n^{\,n/k}(x)\leq \bar a_n^{\,n/k}h^{1-n/k}(x)V^{1/k}(x), \tag{10} \]
where \(V(x)\), \(\bar a_n\) are the same as in (4.1).

The second inequality is a consequence of (7). The first is obtained as follows: in view of (1) and the fact that \(h(x)\leq h(x,v)\),
\[ h_k(x)=h^{1-n/k}(x)\left[\frac{1}{\chi_n}\int_\Omega h^{k-n}(x)h^{-k}(x,v)\,dv\right]^{-1/k} \leq h^{1-n/k}(x)h_n^{\,n/k}(x). \]

Analogously one obtains the lower estimate with \(H(x)=\max h(x,v)\):
\[ h_k(x)\geq H^{1-n/k}(x)h_n^{\,n/k}(x)\geq \beta_n^{\,n/k}H^{1-n/k}(x)V^{1/k}(x). \]

Estimate (10), in particular, determines when and with what rate \(h_k(x)\to0\), if \(x\to x_0\in\partial G\). If at \(x_0\) the body \(G\) is, so to speak, sufficiently convex, then \(V(x)\to0\) so rapidly that also \(h^{1-n/k}(x)V^{1/k}(x)\to0\). For example, from (10) it is easy to conclude that for \(k>n-1\)
\[ h_k(x)\leq Ah^{1-(n-1)/k}(x),\qquad A=A(n,k,V_1). \]
If, however, \(k=n-1\), then \(h_k(x)\) vanishes at all points of \(\partial G\), except those lying inside flat faces.

Further, if \(G\) is contained in a paraboloid of degree \(l\) with vertex \(x_0\in\partial G\), then
\[ h_k(x)\leq Ah^q(x),\qquad q=1-(n-1)(l-1)/kl. \]
If \(G\) contains a similar paraboloid with vertex \(x_0\), then for points on its axis near \(x_0\),
\[ h_k(x)\leq A_1h^q(x). \]
This shows that estimate (10) in this case gives the correct order of magnitude of \(h_k(x)\).

(4.3) If the point \(x_0\in\partial G\) is conical, then
\[ h_0(x)\leq A|x-x_0|^{\omega/\chi_n}, \]
where \(\omega\) is the solid angle of the normals to the supporting planes at \(x_0\). (The proof consists in estimating \(h_0(x)\) for a cone.) If, however, \(x_0\in\partial G\) is the vertex of a paraboloid of degree \(l>1\) contained in \(G\), then \(h_0(x_0)>0\). For \(k<0\), \(h_k(x)>0\) everywhere on \(\partial G\). Let us also note that on the surface of the unit sphere
\[ \ln h_0(x)=(-1)^{n-1}\left(\ln2+\sum_{m=1}^{n-2}\frac{(-1)^m}{m}\right). \]

Received
31 X 1966

REFERENCES

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