Convex Surfaces

CHAPTER I

Extrinsic Geometry

1. Notations, terminology, basic facts

The terminology used in the literature on convex bodies or surfaces is not quite uniform. An agreement, at the outset, in this respect as well as on the notations for certain simple recurring concepts will therefore prove useful.

En is the n-dimensional euclidean space. An r-flat, 0 r n in En is an r-dimensional linear subvariety of En; the 0-, 1- and (n — l)-flats are, of course, also called points, lines, and hyper-planes (planes for n = 3).

In a finite dimensional space like En no actual disagreement on the topology for its r-flats can exist. However, to satisfy readers troubled by the difficulties or ambiguities occurring in infinite-dimensional spaces, we mention that convergence of point sets may in this book always be interpreted in the sense of Hausdorff’s closed limit.¹

If Cartesian coordinates x1, . . . , xn have been introduced in En then a means the point with coordinates a1, . . . , an and | a — b | = [Σ(ai — bi)²]½ is the distance of a and b,

U(p, ) is the open sphere | p — x| < ,

S(p, ) is its boundary | p — x | = .

The closure

of U (p, ) is the solid sphere or ball | p — x | .

Finally, if a and b are two distinct points then the points of the form (1 — ) a + b form for

— ∞ < < ∞      the line L(a, b) through a and b,

0 < ∞         the ray R(a, b) from a through b,

0 1        the (closed) segment E(a, b) from a to b,

0 < 1         the half-open segment E(a, b} from a to b.

With L+(a, b). and correspondingly for the other sets, we mean that orientation of L(a, b) for which increasing yields a traversal in the positive sense.

A non-empty set K in En is convex if it contains with any two points x, y the entire segment E(x, y). Since the intersection of any aggregate of convex sets in En is empty or convex and the r-flats are convex, there is a smallest r-flat, say Vm of dimension m, which contains K. Then K contains m + 1 points which do not lie in an (m — l)-flat, and hence the simplex spanned by these points. Therefore K is a convex set with interior points relative to Vm and K has dimension m, either by definition or in the sense of dimension theory.

The shapes of all convex sets in the euclidean spaces will therefore be known if we know the possible shapes of an n-dimensional convex set K in En. Let p be an interior point of K and U (p, ) ⊂ K. If x is an arbitrary point of K then E (z, x) ⊂ K for any z ∈ U(p, ). Since W = ∪z ∈ ∪(q, p) E(z, x} is open, every point of E (p, x) is an interior point of K. Therefore:

(1.1) The interior points of K form a convex set.

(1.2) The closure of K is a convex.

It also follows thatfor every y ∈ S(p, ) the ray R (p, y) either lies entirely in K or intersects the boundary B(K) of K in exactly one point b(y). In the latter case all points of R+(p, b(y)) following b(y) lie outside K and all points of E (p, b(y)} are interior points of K.

We put δ(y) = | p — b(y) | and δ(y = ∞ if R(p, y) ⊂ K. It follows easily from W ⊂ K for x ∈ K that δ(y) is a continuous function of y. Hence

(1.3) The interior of an n-dimensional convex set K is homeo-morphic to En.

If K is bounded then its boundary B(K) is homeomorphic to the (n — 1)-dimensional sphere Sn−1. When K is not bounded we consider first the case where K does not contain a complete line. The set M of those y on S(p, ) for which δ(y) = ∞ is then nonempty and closed, and contains no two antipodal points of S(p, ). The union of all rays R (p, y) ⊂ K is evidently a convex, possibly degenerate, cone with apex p, and the intersection of this cone with is a bounded closed convex set. Therefore S(p, ) — M and hence B(K) is homeomorphic to En−1.

If K contains a complete line then it contains a flat Vr of maximal dimension r. If r = n then K = En. Assume 1 r n — 1 and take any (n — r)-flat Vn−r normal to Vr. Then K′ = Vn−r ∩ K is an (n — r)-dimensional convex set in Vn−r which does not contain a complete line, so that its boundary B(K′) is by the preceding discussion homeomorphic to Sn−r−1 or to En−r−1. Since K contains for q ∈ K′ and x ∈ Vr the segment E(q, x), the closure of K contains with any q ∈ K′ the r-flat through q parallel to Vr. Therefore B(K) consists of the r-flats parallel to Vr through points of B(K′) and is therefore homeomorphic either to the product En−r−1 × Er, i.e., to En−1 or to Sn−r−1 × Er, which may be regarded as a cylinder with r-dimensional generators.

(1.4) If the boundary B(K) of an n-dimensional convex set K in Enis not empty (K ≠ En) then it is homeomorphic either to En−1 or to Sn−1 or to the product Sn−r−1 × Er, 1 r n — 1.

The product Sn−r−1 × Er is connected except for r = n — 1, where B(K) consists of two parallel hyperplanes.

A (complete) convex hypersurface in En is the boundary of an w-dimensional convex set K in En provided it is non-empty and connected, i.e., B(K) is a convex hypersurface unless K is the whole space or a set bounded by two parallel hyperplanes. Since convex surfaces are the primary subject of this book we reformulate (1.4):

(1.5) A (complete) convex hypersurface in En is either homeomorphic to Sn−1 or to En−1, or to a product Sn−r−1 × Er, 1 r n — 2 and is, respectively, called closed or open or cylindrical.

For n = 2, 3 we use, of course, the terms curve and surface instead of hypersurface. A (complete) convex curve is homeo-morphic to a line or to a circle, and a (complete) convex surface is homeomorphic to a plane, a two-sphere or to an ordinary cylinder. The term convex hypersurface will also be used for relative open connected subsets of complete convex hyper surf aces. The parentheses indicate that complete will be omitted when the distinction is inessential or clear from the context.

A supporting r-flat of the n-dimensional convex set K in En is an r-flat which contains points of but no interior points of K. The supporting 0-flats are the points of B(K), and the supporting (n — l)-flats, simply called supporting planes, are the hyperplanes which contain points of K but do not separate any two points of K. This property is used as definition for supporting planes of a convex set without interior points, but for obvious reasons supporting r-flats with r < n — 1 are not defined for such sets. Notice the following obvious, but useful, fact:

(1.6) If V1 V2 . . . are supporting r-flats of K and tend to an r-flat V which contains points of (when K is bounded lim Vn automatically contains points of ), then V is a supporting r-flat of K.

Next we show

(1.7) If K1 is a bounded convex set and K2 any convex set and then points exist such that

If P1, P2, P are the hyperplanes normal to L (f1, f2) at f1, f2 and (f1+f2)/2 then Pi is a supporting plane of Ki and P separates K1 from K2. If K1 is a point f1 then f2 is unique.

The existence of f1 and f2 follows from the boundedness of K1. If P2 were not a supporting plane of K2 then there would be a point q of K2 on the same side of P2 as f1. But the angle is acute, hence | f1 — (1 — ) f2 — q | < | f2 — f1| for small positive and (1 — ) f2 + q ∈ K2. It is clear that P separates K1 from K2 and has, in fact, distance | f1 — f2 |/2 from Ki.

If K1 = f1 then f2 is called a foot of f1 on K2. A second foot g of f1 on K2 would lead to the contradiction (f2 + g)/2 ∈ K2 and | f1 — (f2 + g)/2 | < | f1 — f2 | = |f1 — g |.

We use (1.7) to prove:

(1.8) If an arbitrary convex set K′ and an n-dimensional convexset K in En have a common boundary point p and K′ contains nointerior points of K, then K and K′ possess at p a common supporting plane. Consequently, a supporting r-flat of K, in particular a pointof B(K), lies in a supporting plane of K.

Choose any interior point q of K and call Kn the set obtained from by the dilation with center q and factor 1 — n−1. If then K′n∩Kn = 0 and (1.7) furnishes a hyperplane Pn which separates K′n from Kn and contains, therefore, a point of the segment E(p, pn) from p to the image pn of p under the dilation. Since | p — pn | = | p — q|/n, there is a subsequence {Pv} of {Pn} which tends to a hyperplane P ⊃ p. Then P cannot separate two points a1, a2 of K because there are points aiv ∈ Kv which tend to ai and Pv would separate a1v from a2v for large υ. Similarly, P does not separate two points a′1 and a′2 of K′, because a′i ∈ K′n for large n, and Pv does not separate a′1 from a′2 for v > n.

A supporting half space of the convex set K is a closed half space bounded by a supporting plane of K and containing K. A corollary of (1.8) is:

(1.9) If K ≠ En then is the intersection of the supportinghalf spaces of K.

Observe the further corollary of (1.8):

(1.10) If av and bv, υ = 1, 2, . . ., are two distinct points on the boundary B(K) of the n-dimensional set K which tend to the same point p, then the limit L of any converging subsequence of L(av ,bv) lies in a supporting plane of K at p.

For L is a supporting line of K because all interior points of K on L(av, bv) must lie on E(av, bv).

Let C be a convex hypersurface bounding the convex set K, which is uniquely determined by C except in the trivial case where C is a hyperplane. The interior of K and the supporting planes of K are also called the interior and the supporting planes of C.

The intersection of all supporting half spaces of K bounded by supporting planes of K at a fixed point b of C is a convex cone T0 with interior points whose boundary T is, by definition, the tangent cone of K or C at b. If T is a hyperplane, then it is the unique supporting plane of C at b. We then also say that T is the tangent plane of C at b and that C is differentiable at b. We conclude from (1.6):

(1.11) If C is differentiable at the points x ∈ M ⊂ C then the tangent plane of C at x depends continuously on x.

A convex hypersurface C is locally representable in the form z = f(x1, . . ., xn−1). This is quite obvious if we are satisfied with oblique coordinates. It suffices to choose an arbitrary supporting plane of C at a given point b as z = 0 and any line L through b and an interior point of C as z-axis. Then (1.10) shows that a line parallel and close to L intersects C exactly once in a neighborhood of b. To see that rectangular coordinates with this property exist, we form the complementary or polar cone T′ to T which consists of the rays normal at b to the supporting plane of T and in the exterior of T. (If T is a plane then T′ consists of the ray normal to T and in the exterior of K.) It is easy to see that T′ is convex, see K. p. 4. Because T0 ⊃ K the cone T′ possesses at b a supporting plane P touching T′ only at b. The line L normal to P contains interior points of K, with P as plane z = 0 and L pointing into the interior of K as positive z-axis we have a rectangular coordinate system which satisfies the following theorem:

(1.12) THEOREM. For a given point b on a convex hypersurface C there is a neighborhood Cb of b on C and a system of rectangular coordinates x1, x2, . . ., xn−1, z with b as origin such that Cb is representable in the form

.

Moreover f(x) is a convex function of x, ² and its difference quotients | f(x) — f(y) |/| x — y | , x ≠ y, are bounded.

If (x0 , z0) is a point of Cb where C (or C0) is differentiable then f(x) possesses at x0 a differential.

The convexity of f(x) follows at once from the convexity of the set | x | δ, z f(x) in En. The boundedness of the difference quotients is a consequence of (1.10), because the z-axis contains interior points of K, and so is the existence of the differential. For if z = 0 is the tangent plane of C at b, then (1.10) implies the much stronger property that for a given ∈ > 0 and a suitable ∈′ > 0 the inequality

.

The boundedness of the difference quotients shows that we need not worry regarding the concept of area of a convex hypersurface. Also, a set of measure 0 on has a projection of measure 0 in | x | δ.

2. Convex curves

In this and the next section we discuss the differentiability properties of convex hypersurfaces in greater detail. Since these are local we may consider partial surfaces representable in the form (1.12).

For a function f(x) of one variable x, a x b, we use

as symbols for the right and left derivatives.

If y = f{x) is a convex curve, then for x1 < x2, < x′1 < x′2, x1 x′1, and x2 x′2

hence f′ r(x) and f′ i(x) exist; and if f′m(x) is the slope of an arbitrary supporting line at x, then

.

The monotoneity of f′m(x) implies that the number of points where f′(x) does not exist is, at most, denumerable. It also implies that f′m(x) has almost everywhere a derivative, and that all points where the derivatives of f(x) and f′m(x) exist, the latter has the same value for different choices of f′m(x). We therefore simply denote it as the (finite) second derivative f″(x) of f(x) at x. We remember for later purposes that:

We now investigate the curvature of convex curves. The radius of the circle (possibly ∞) which passes through the points (x0, y0) and (x0 + h, y0 + k) and has slope m at (x0, y0) equals

Its center lies, of course, on the line (x — x0) = —m(y — y0). A line which passes through (x0 + h, y0 + k) and is normal to the line with slope mh intersects x = — my at a point whose distance from (x0, y0) equals

Now consider a continuous curve y = f(x) which has for x = x0 a derivative. The radius rh(x0) of the circle which has the same tangent as the curve at (x0, y0 = f(x0)) = p0 and passes through ph = (x0 + h, f(x0 + h) = y0 + k), h > 0, has by (2.3) a right limit if the so-called second right de la Vallée-Poussin derivative

exists and

If f(x) possesses a derivative in a neighborhood of x0 then the distance Rh(x0) from p0 of the intersection of the normals to y = f(x) at p0 and ph has a limit if f′(x) possesses at x0 the right derivative

and

.

It follows from Taylor’s theorem that the existence of f″r(x0) implies the existence of D″r f(x0) and that then the two derivatives are equal. The function y = x³ sin x—1 shows that the converse need not hold; in fact D″f(x0) = D″r f(x0) = D″i f(x0) may exist in a set of positive measure without f″(x0) existing anywhere, see Denjoy [1].

If, for any differentiable f(x), the circle of radius rh(x0) touching y = f(x) at p0 and passing through ph has center ah, then the arc contains a point ph′, 0 < h′ < h whose distance from ah is maximal or minimal. The normal to the curve at this point passes through ah so that Rh′(x0) = rh(x0). Hence, and similarly,

If f(x) is a convex function and f′(x0) exists, then we will interpret f′(x0 + h) or mh in (2.4) as the slope f′m(x0 + h) of any supporting line at ph. Because f′m(x0 + h) is monotone and are independent of the choice of f′m(x0 + h). Then (2.9) holds and, if

,

exists for one choice of f′m(x0 + h), it exists for all and is independent of the choice.

Jessen [1] discovered the surprising fact that for convex f(x) the existence of D″r f(x0) and that of f″r(x0) are equivalent. He deduced this from the following more general result:

(2.10) THEOREM. Let the convex curve y = f(x) possess a tangent at [x0, f(x0)] and put

.

Then

,

where for rs = ∞ the left member means ri/2.

For a proof we consider the two circles

A: x² + (y — α)² = α², B: x² + (y — β)² = β², α > β > 0.

From p1 = (α, α) ∈ A we draw the tangent to B which has the greater slope and call p2 its second intersection with A. We proceed with p2 as with p1 i.e., we draw the tangent different from L(p1, p2) to B; it intersects A a second time at p3. Continuing in this manner we obtain an infinite polygon ∪E(pi, pi+1) which together with the arc — α < x < 0 of A forms a convex curve D of the form y = f(x) which has at p0 = (0,0) the x-axis as tangent. Clearly for this curve

If pi = (hi, ki) and δi is the acute angle formed by L(pi—1, pi) and the x-axis, then the minimum and maximum for Rh(0) on the interval hi < h < hi−1 are reached at the end points, hence from (2.4)

because | pi | hi—1 = (1 + ki² hi—2)½ → 1.

Let qi = (h′i, k′i) be the foot of the