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ISOPERIMETRIC INEQUALITIES FOR CONVEX PLANE CURVES. Mark Green* and Stanley Osher**. U.C.L.A.. INTRODUCTION. In this paper, we obtain someĀ ...
STEINER POLYNOMIALS, WULFF FLOWS, AND SOME NEW ISOPERIMETRIC INEQUALITIES FOR CONVEX PLANE CURVES Mark Green* and Stanley Osher** U.C.L.A. INTRODUCTION In this paper, we obtain some new inequalities for integrals of convex functions of the curvature (resp. Wul curvature) of convex plane curves. We also show that the di erence between the two sides of our inequalities are monotone decreasing as the region ows under the unit-speed outward normal (resp. Wul ) ow. Given a bounded plane region K , the unit-speed outward normal ow has been highly studied, and is of interest in many applied problems, e.g. combustion. If instead K grows by varying the outward normal speed to be a function () of the direction of the unit normal, one has the Wul ow, which is also of considerable interest, e.g. in studying the growth of crystals [O-M]. When the region K is convex, there is a simple closed-form expression which describes these ows, and the region converges to a disk in the rst case and a Wul shape in the second. The area of the region when the initial region K is convex is a polynomial in t, known respectively as the Steiner polynomial or Wul -Steiner polynomial. A novel feature of our approach is that we study the roots of the Steiner and Wul Steiner polynomials, which occur at negative values of t. The classical isoperimetric inequality in both cases states that these polynomials have (negative) real roots t1  t2, and that they are distinct if and only if K is not a disc (respectively not a Wul shape). Bonnesen's inequality states that the inradius and outradius ri and re lie in the interval [?t1; ?t2], and in the open interval if K is not a disk (respectively not a Wul shape). Our inequalities are most naturally stated and proved in terms of the roots t1 and t2. We feel that this is a potentially quite fruitful approach to studying convex bodies in higher dimensions. In the context of this new approach, a very natural link between the outward normal and Wul ows and the curvature integrals of the region appears. In important cases, the quantities that our inequalities state are positive are shown to be monotone decreasing as the region evolves under the ow. Particularly suggestive is the fact that the entropy of the curvature (respectively Wul curvature) is bounded above in terms of the area and is monotone decreasing with time. The inequalities themselves are quite fascinating. It came as a surprise to us that there are interesting new things to be said about convex plane curves. We state our inequalities here for arbitrary smooth bounded convex plane regions K in the curvature case, and leave the Wul case to the body of the paper. One of them is due to Gage [G], whose result was * Research partially supported by NSF-DMS-9401256 ** Research partially supported by the NSF and DARPA through NSF-DMS-9615854 and NSF-DMS-9706827 1

a source of inspiration to us. Gage's result is Z

@K

k2ds  L A;

which he points out fails for non-convex curves. For m  3, the inequalities appear to be new. For example, Z 2 2 k3ds  L  ?A22 A  @K

and Z

3 2 k4ds  L  ?A33AL : @K

The general result is: THEOREM 0.1. Let K be a bounded convex plane region with radius of curvature () viewed as a function of the direction of the unit normal vector ~n(), and let t1  t2 be the roots of the Steiner polynomial of K . Let F (x) be a convex function on R+, i.e. F 00 (x)  0. Then 1 Z F ()d  1 (F (?t ) + F (?t )); 1 2 2 1 2 S

where t1, t2 are the roots of the Steiner polynomial of K . Two especially interesting cases of our inequalities are:

THEOREM 0.2 (Entropy inequality for curvature). Z

@K

Equality holds if K is a disk. THEOREM 0.3. Let Then

Z

@K

p

klog(k A=)ds  0:

r

e = 1 ? 4LA 2 : p 1 + e )  0: log( log(k A= )ds + Le 2 1?e

Equality holds if K is a disk. We also obtain versions of these theorems for the Wul case (see de nition 1.13 below). For example, we have: 2

THEOREM 0.4. For K , W bounded convex sets, W symmetric, with support functions p, respectively, and Wul radius of curvature p00 ; W = p + + 00 we have for any convex function F , 1 Z F ( ) ( + 00 )d  1 (F (?t ) + F (?t )); 1 2 2AW S1 W 2 where t1; t2 are the roots of the Wul -Steiner polynomial. We also show: THEOREM 0.5. If in addition to the other hypotheses, F 0 (x) is concave, the di erence between the two sides of the inequality in Theorems 0.1 , 0.2, and 0.4 is monotone decreasing and for 0.3 is monotone increasing under the unit speed outward normal ow and Wul ows respectively. We note in particular the following physically intriguing result: THEOREM 0.6. The entropy ?

Z

@K

p

klog(k A= )ds

is increasing under the unit-speed outward normal ow. An unusual feature of Theorem 0.1 is that its most natural statement involves the roots of the Steiner polynomial, which would appear to have no direct geometric signi cance, since they appear in the negative range of t|yet the roots can be eliminated from any given case and replaced by a function of L and A. The proof of Theorem 0.1 involves three steps, each one a bit non-standard. The rst step is an interesting \two-piece" version Jensen's inequality (Proposition 2.6). The second step is a bound Rfor ?t2, i.e. for one of the roots of the Steiner polynomial, in terms of the largest integral I ()d over a subset I of [0; 2] of measure  (Proposition 2.7). This inequality can be obtained for symmetric K with some work from Bonnesen's inequality. The third step is a somewhat unusual symmetrization argument (Proposition 2.8). Theorems 0.1-0.3 and 0.5-0.6 all generalize to the Wul case. This is done in section 3. For example, one has: THEOREM 0.7. For W bounded convex symmetric with support function and K bounded convex with Wul curvature kW , Z kW2 ds  AAW L (Wul Gage inequality); K @K Z A L2 ? 2A2 A kW3 ds  W A2 W K ; @K K Z p kW log(kW AK =AW ) ds  0 (Wul entropy inequality) : @K

3

In all three cases, the di erence between the two sides of the inequality is monotone decreasing under the outward Wul ow. 1. CONVEX PLANE CURVES, STEINER POLYNOMIALS, AND WULFF FLOWS In this section, we collect together some very classical facts about convex plane curves that exist in disparate sources (see [Sc] for an encylopaedic survey). We present them here from a novel perspective, in which the roots of the Steiner and Wul -Steiner polynomials play a central role. DEFINITION 1.1. If K  Rn is a bounded convex region, then

tK = ftp j p 2 K g and if K1; K2  Rn are two bounded convex region, then

K1 + K2 = fp + q j p 2 K1 ; q 2 K2g: We have the following well-known fact: PROPOSITION 1.2. For a bounded convex region K , the region inside the image of @K under the constant speed outward ow at time t  0 is K + tB, where B denotes the unit ball. DEFINITION 1.3. The support function p of a bounded convex region K is de ned by p(~v ) = maxf~x  ~v j ~x 2 K g: It is elementary that: PROPOSITION 1.4. If K1 , K2 are bounded convex regions with support functions p1; p2 , then the support function of t1K1 + t2K2 is t1 p1 + t2p2 . PROPOSITION 1.5. Let W be a bounded convex region having support function . The image of a bounded convex region K at time t  0 under the normal ow having speed (~n) (the Wul ow associated to W ) is K + tW . We now restrict ourselves to the case of bounded convex regions in the plane. PROPOSITION 1.6. Let K be a bounded convex region in the plane having area A and perimeter L, and where @K has element of arc-length ds and curvature k. We have the following formulas: (1) ds = (p + p00 )d; (2) k = 1=(p +Rp00 ); (3) A = R(1=2) S1 p(p + p00R)d; (4) L = S1 (p + p00 )d = S1 pd; 4

(5) If ~r() is the position vector of the point of @K whose normal vector is ~n() = (cos ; sin ), then ~r() = p()~n() + p0 ()~n0 (): PROOF: We start from the basic fact

p() = ~r()  ~n(): Di erentiating, we get

p0 () = ~r0 ()  ~n() + ~r()  ~n0 () = ~r()  ~n0 (): It follows that

~r() = p()~n() + p0 ()~n0 (); proving (5). Di erentiating, we get ~r0 () = p0 ()~n() + p()~n0 () + p00 ()~n0 () + p0 ()~n00 (): Since

~n00 () = ?~n();

we have

~r0 () = (p() + p00 ())~n0 ():

Taking lengths, we get

ds = p + p00 ; d R proving (1) and (2). The rst part of (4) follows from (1), but then S1 p00 d = 0, giving the second formula. Now the element of area is given by dA = (1=2)~r() ^ ~r0 ()d = (1=2)p(p + p00 )~n ^ ~n0 d = (1=2)p(p + p00 )d; from which (3) follows. COROLLARY 1.7. Let K be a bounded plane region with area A and perimeter L. Let AK (t) denote the area of K + tB. Then for all t  0, AK (t) = A + Lt + t2: PROOF: By Proposition 1.4, the support function for K + tB is p + t. It follows that AK (t) = (1=2) = (1=2)

Z

Z

S1 S1

(p + t)(p + t + p00 )d

p(p + p00 )d + (1=2)t

= A + Lt + t2 : 5

Z

S1

(2p + p00 )d + (1=2)t2

Z

S1

d

REMARK: This polynomial is called the Steiner polynomial of K . A remarkable fact is that the behavior of the Steiner polynomial of K for negative values of t is quite interesting. For example, we have that 2 min(AK (t)) = A(? 2L ) = A ? L4 : The isoperimetric inequality is thus equivalent to saying that min(AK (t))  0; with equality if and only if K is a disc. DEFINITION 1.8. Let t1  t2 be the roots of AK (t). Let ri and re be the radii of the largest inscribed and smallest circumscribed circles of K respectively (called the inradius and outradius). Let k be the curvature of @K ,  = k1 the radius of curvature, and min and max the minimum and maximum values of . These quantities are all equal if K is a disc. REMARK 1.9: For the constant speed outward ow,

AK+tB (s) = AK (t + s) and thus If follows that We have the formulae

ti (K + tB) = ti(K ) ? t; i = 1; 2: d (t (K + tB)) = ?1 i = 1; 2: dt i

p

2 t1 = ? 2L + L 2? 4A ; p 2 L t2 = ? 2 ? L 2? 4A : The fact that the roots are real is equivalent to the isoperimetric inequality. THEOREM 1.10. If K is strictly convex and is not a disc, then ?max < t2 < ?re < ? 2L < ?ri < t1 < ?min  0: In particular, A(?min ) > 0; A(?max) > 0 while if ?re  t  ?ri , A(?t) < 0: REMARK: The last inequality more explicitly states that when ?re  t  ?ri , A ? Lt + t2 < 0:

6

This is known as Bonnesen's inequality. PROOF: If we follow a given point inward under the constant speed normal ow, the time of rst shock is ?. Thus up to t = ?min, the region remains convex and of positive area. Unless the region is a disk, the area remains positive even at t = ?min. This proves

t1 < ?min: To prove Bonnesen's inequality, we rst note the following lemma: LEMMA 1.11. Let C be either a maximal inscribed or minimal circumscribed circle for K . Then every closed semi-circle contains a point where C is tangent to @K . PROOF OF LEMMA: We do the case of the inscribed circle, the other being similar. Let v , 2 A be the unit vectors going from the center of C to the points of tangency. If the conclusion fails, then there exists a unit vector v such that v  v < 0 for all |take v pointing toward the midpoint of the semicircle that contains no v . Now if we move the center of C by v for  > 0 small enough, then C continues to lie in the interior of K and has no points of tangency. So we may slightly increase the radius of C , contradicting maximality. This proves the lemma. Returning to the proof of Bonnesen's theorem, choose points of tangency 1 ; : : : ; N on C so that every half-circle contains one. Now Z 1 A(?ri ) = 2 1 (p ? ri )(p + p00 ? ri )d: S We now use the well-known: LEMMA 1.12 (Poincare). Let f be a function on [0; a] whose rst derivative is in L2 and such that f (0) = f (a) = 0: Then Z Z  2 0 2 f (x) dx  ( a ) f (x)2 dx; [0;a] [0;a] with equality i f = A cos (x=a) + B sin (x=a). In particular, if a  , Z

[0;a]

Equality holds i

f 0 (x)2 dx 

Z

[0;a]

f (x)2 dx:

f (x) = c sin(x=a)

for some constant c. PROOF: By rescaling, we may take a = . Extend f to be an odd function on [?; ] and then extend to be periodic with period 2. Now f has rst derivative in L2, and if we expand by Fourier series X f (x) = an einx; n2Z

7

then a0 = 0 since f is odd. Now Z

[0;]

(f 0 (x)2 ? f (x)2 )dx = 

X

(n2 ? 1)jan j2:

n6=0

This gives the inequality since n2  1 for n 6= 0, and equality holds i an = 0 for n 6= 1, and then from the boundary conditions f (x) = c sin(x). Given this Lemma, by integration by parts we have Z 1 A(?ri ) = 2 1 ((p ? ri )2 ? (p0 )2 )d: S

We can break up the right-hand side into integrals over the intervals [i ; i+1 ], and then apply the Lemma to see that A(?ri )  0. For equality to hold, we would need p = ri + a cos  + b sin , which gives a circle of radius ri with center (?a; ?b). The same argument works for re , and thus

t2 < ?re < ?ri < t1: If we choose the origin to be the center of a maximal inscribed circle, then for all , ri  p(), integrating over the circle gives 2ri 

Z

S1

pd = L

and it is easy to see we have equality only for a circle. A similar argument (with a di erent origin) gives L  2re : So ?re < ? 2L < ?ri :

It remains to see that ?max < t2. If we look at the curve whose support function is p ? t for t  max, it is once again a convex curve (all points have gone past their shock, i.e. p ? max  0), and its area is A(?t), so we see that

?max < ?t2: This completes the proof. All of this is equally valid in the Wul case. DEFINITION 1.13. Let K; W be bounded convex regions in the plane having areas AK , AW and having support functions p; respectively. The Wul length of @K with respect to W is Z L =

ds: @K

8

The Wul curvature is given by + 00 kW = p + p00 and its inverse the Wul radius of curvature is given by

p00 : W = p + + 00 Let AK;W (t) denote the area of K + tW .

PROPOSITION 1.14.

AK;W (t) = AK + L t + AW t2 : PROOF: By Proposition 1.4, the support function of K + tW is p + t . It follows that

AK;W (t) = (1=2)

Z

S1

(p + t )(p + p00 + t( + 00 ))d;

and the proposition follows from expanding this in t. DEFINITION 1.15. Let t1  t2 be the roots of AK;W (t). Let

ri = maxft j some translate of tW is contained in K g and

re = minft j some translate of tW contains K g (called the W-inradius and W-outradius). Let kW be the Wul curvature of @K , W = k1W the Wul radius of curvature, and min and max the minimumand maximum values of W . These quantities are all equal if K = tW + ~a for some tand ~a. THEOREM 1.16. For K , W strictly convex, if K 6= tW + ~a for any t and ~a, then ?max < t2 < ?re < ? 2LA < ?ri < t1 < ?min  0: W In particular, AK;W (?min ) > 0; AK;W (?max) > 0 while if ?re  t  ?ri , AK;W (?t) < 0: The proof is essentially the same as the previous theorem. There are a large number of classical results which are all consequences of this theorem, e.g. see pp. 119-121 of [S]. We note in particular: 9

COROLLARY 1.17 (Wul -Bonnesen inequality). If K 6= sW + ~a for any s and ~a

and ri  t  re , then

AK ? L t + AW t2 < 0: PROPOSITION 1.18. For a convex bounded plane region K , the isoperimetric defect L2 ? 4A is an invariant of the outward unit-speed ow. For K; W bounded convex plane regions, the Wul isoperimetric defect L2 ? 4AW A is an invariant of the outward Wul ow. PROOF: This is an easy direct computation. However, the elegant way is to notice that these quantities can be computed from the minima of AK (t) and AK;W (t), and therefore are invariants of the ow. Under the outward normal and Wul ows, a region converges to a ball or Wul shape. We will make crucial use of this fact in the second part of the paper. PROPOSITION 1.19. Let K; W be bounded convex regions in the plane. Then as t ! 1, the region (1=t)(K + tW ) converges to W in the strong sense that their support functions converge. PROOF: The support function of (1=t)(K + tW ) is (p=t) + , and this converges to as t ! 1. PROPOSITION 1.20. Under the unit-speed outward normal ow, the curvature k satis es dk = ?k2: d = 1; dt dt Under the Wul ow for the region W with support function , the Wul curvature kW satis es dW = 1; dkW = ?k2 : W dt dt PROOF: The support function for K + tW is p + t . Since W = (p + p00 )=( + 00 ), we have that dW = ( + 00 ) dt

+ 00 = 1: Now dkW = d?W1 = ?k2 : W dt dt Taking = 1 gives the result for the constant speed outward ow. 2. SOME NEW INEQUALITIES OF ISOPERIMETRIC TYPE FOR CONVEX PLANE CURVES Throughout this section, let K be a bounded convex plane region whose boundary has curvature k and radius of curvature . Let t1, t2 denote the roots of the Steiner polynomial of K . 10

THEOREM 2.1. Let F (x) be a convex function on R+, i.e. F 00 (x)  0. Then

1 Z F ()d  1 (F (?t ) + F (?t )): 1 2 2 S1 2 REMARK: We may rewrite this in the more geometrically appealing form 1 Z kF ( 1 )ds = 1 Z F () ds  1 (F (?t ) + F (?t )): 1 2 2 S1 k 2 S1  2 For F (x) = 1=x, this is the Gage inequality [G] Z k2ds  L A; @K which he points out fails for non-convex curves. For most other F 's, the inequalities appear to be new. For example, Z 2 2 k3ds  L  ?A22 A  @K

and

Z

3 2 k4ds  L  ?A33AL : @K Before coming to the proof, we state two further special cases:

THEOREM 2.2 (Entropy inequality for curvature). Z

@K

klog(k)ds + log( A )  0:

Equality holds if K is a disk. REMARK: Note that either term in the inequality can be negative. An alternate formulation is Z p klog(k A=)ds  0: @K

THEOREM 2.3. Let Then

r

e = 1 ? 4LA 2 : Z

@K

p 1 + e )  0: log(k A= )ds + Le log( 2 1?e

Equality holds if K is a disk. REMARK: This may be stated more compactly as Z + e )e=2 kpA=) ds  0: log(( 11 ? e @K We also have the following result: 11

THEOREM 2.4. Let F (x) be a C 3 convex function on R+ such that F 0 (x) is concave, then the di erence

1 Z F (())d ? 1 (F (?t ) + F (?t )) 1 2 2 S1 2 is monotone decreasing under the unit-speed outward normal ow. PROOF: Under the unit-speed outward normal ow, d = 1; dti = ?1 dt dt for i = 1; 2. The derivative of the di erence in the theorem is 1 Z F 0 (())d ? 1 (F 0 (?t ) + F 0 (?t )): 1 2 2 S1 2 Since ?F 0 (x) satis es the hypotheses of the Theorem 2.1, this quantity is  0. As a preliminary to proving the main theorem, we make the following de nition and notation: DEFINITION 2.5. Consider Z Z 1 supf ()d j I  S ; d = g: I

I

Let I1 denote a subset of S 1 of measure  realizing this bound, and let I2 be its complement. There exists a real number a such that I1  f j ()  ag and I2  f j ()  ag: We set Z 1 1 =  ()d

and

We note that

I1

Z 1 2 =  ()d: I2

1 + 2 = L : PROPOSITION 2.6. Let F (x) be a convex function on R+ . Then 1 Z F (())d  1 (F ( ) + F ( )): 1 2 2 S1 2 PROOF: By Jensen's theorem, applied to Ii, 1 Z F (())d  F ( 1 Z ()d) = F ( ): i  Ii  Ii Taking the average of these two inequalities for i = 1; 2 gives the desired result. Recall that a convex region is called symmetric if p() = p( + ) for all . 12

PROPOSITION 2.7. For K a symmetric bounded convex region, 1  ?t2: PROOF: We rst note that for a symmetric region K , if an inscribed circle of radius ri has center C , then a circle of radius ri and center ?C is also inscribed. By convexity, the circle with radius ri and centered at the origin is also inside K , and therefore p()  ri for all . By a similar argument, p()  re for all . By the results of section 1, ?t1  p()  ?t2 for all . We recall that ?t1 = 2L ? u; ?t2 = 2L + u; where p 2 u = L 2? 4A : Thus ?u  p() ? 2L  u for all . Since 1 + 2 = L=, we may write 1 = 2L + b; 2 = 2L ? b for some b  0. The inequality we are trying to prove, 1  ?t2; is equivalent to proving b  u: On I1, () ? a  0; so we have ?(p() ? 2L )(() ? a)  u(() ? a): Integrating, we have Z 1 ?  (p() ? 2L )(() ? a)d  u(1 ? a): I1 13

On I2, so we have Integrating, we have

() ? a < 0;

?(p() ? 2L )(() ? a)  ?u(() ? a): Z 1 ?  (p() ? 2L )(() ? a)d  ?u(2 ? a): I2

Adding these two inequalities gives Z 1 ?  1 (p() ? 2L )(() ? a)d  u(1 ? 2): S

The left-hand side simpli es to 2 (L2 ? 4A) = 2u2: 42 The right-hand side is 2ub. The inequality is thus 2u2  2ub and thus as desired.

ub

PROPOSITION 2.8. For K a bounded convex region (not necessarily symmetric), 1  ?t2: PROOF: We proceed by a symmetrization argument. Given K , for any  we divide K into two regions by joining the points on @K corresponding to ,  +  by a straight line. Let L1, L2 be the lengths of the two pieces of @K and A1 , A2 the areas of the two halves; thus

L = L 1 + L 2 ; A = A 1 + A2 : Choose  so that

(2L1)2 ? 8A1 = (2L2 )2 ? 8A2 ; this is possible because these quantities vary continuously with  and are interchanged as  runs from 0 to , so by the intermediate value theorem their di erence must go through 0. Let K1 be the symmetric convex body obtained by joining the rst half of K to a copy of itself rotated by 180 degrees, and K2 a symmetric convex body obtained by doing the same thing to the second half. Thus Ki has perimeter 2Li and area 2Ai for i = 1; 2. 14

Now for symmetric convex regions, we may take the subset I1 of the circle to be R symmetric. Since I ()d is maximized by I1 among all subsets of measure , it follows that Z Z 1 ()d + ()d) = 21 (1(K1 ) + 1(K2 )): 1(K )  2 ( I1 (K1 )

I1 (K2 )

Now by Proposition 2.7 applied to the symmetric regions K1 , K2 , p

2 1(K1 )  22L1 + (2L1)2? 8A1

and

p

2 1(K2 )  22L2 + (2L2 )2? 8A2 :

Taking the average,

p

p

2 2 1 (K )  L1 2+ L2 + 21 ( (2L1)2? 8A1 + (2L2 )2? 8A2 ):

We now use the way the decomposition was chosen to recall that (2L1)2 ? 8A1 = (2L2 )2 ? 8A2 : Call this quantity , so The inequality we want is

p

1 (K )  2L + 2 : p

2 1(K )  2L + (L1 + L2 ) 2? 4(A1 + A2 ) :

We will be done if we can show that

 (L1 + L2 )2 ? 4(A1 + A2 ): Now

= 2L21 ? 4A1 + 2L22 ? 4A2 = (L1 + L2)2 + (L1 ? L2)2 ? 4(A1 + A2 )  (L1 + L2)2 ? 4(A1 + A2): This proves the proposition. We have the elementary lemma: 15

LEMMA 2.9. If F (x) is convex, then if b  a  0 and c arbitrary, then

F (c ? a) + F (c + a)  F (c ? b) + F (c + b): PROOF: By the mean value theorem, F (c ? a) ? F (c ? b) = (b ? a)F 0 (e1 ) for some c ? b  e1  c ? a and F (c + b) ? F (c + a) = (b ? a)F 0 (e2 ) for some c + a  e2  c + b. Thus e2  e1 , and since F 0 (x) is increasing, F (c ? a) ? F (c ? b)  F (c + b) ? F (c + a); and the Lemma follows. PROOF OF THEOREM 2.1: We already have that 1 Z F (())d  1 (F ( ) + F ( )): 1 2 2 S1 2 Now 1 = 2L + b; 2 = 2L ? b and ?t2 = 2L + u; ?t1 = 2L ? u: By Proposition 2.8, b  u, and so by Lemma 2.9, F (1 ) + F (2 )  F (?t1 ) + F (?t2): The Theorem follows by combining these two inequalities. 3. ISOPERIMETRIC INEQUALITIES IN THE WULFF CASE Here the main result is: THEOREM 3.1. For K , W bounded convex sets, W symmetric, with support functions p, respectively,and Wul radius of curvature W , we have for any convex F (x), 1 Z F ( ) ( + 00 )d  1 (F (?t ) + F (?t )); 1 2 2AW S1 W 2 where t1; t2 are the roots of the Wul -Steiner polynomial. REMARK: One may re-write this as 1 Z k F ( 1 ) ds  1 (F (?t ) + F (?t )) 1 2 2AW S1 W kW 2 where kW is the Wul curvature. We also have: 16

THEOREM 3.2. Let F (x) be a C 3 convex function on R+ such that F 0 (x) is concave. For K convex and W convex and symmetric, the di erence 1 2AW

Z

S

F (W ) ( + 00 )d ? 21 (F (?t1 ) + F (?t2 )) 1

is monotone decreasing under the outward Wul ow. PROOF: The argument is the same as for Theorem 2.4, once we note that

dW = 1 dt and

dti = ?1 dt

for i = 1; 2. PROOF OF THEOREM 3.1: The argument proceeds along very similar lines to that of section 2, but with some modi cations. DEFINITION 3.3. Consider Z

supf

I

W ( + 00 )d j I  S 1;

Z

I

( + 00 )d = AW g:

Let I1 denote a subset of S 1 realizing this bound, and let I2 be its complement. There exists a real number a such that

I1  f j W ()  ag and

I2  f j W ()  ag:

We set and We note that

1 = A1 W

Z

1 = A1 W

Z

I1

I2

W ( + 00 )d W ( + 00 )d:

1 + 2 = AL : W

By Jensen's Theorem, we obtain by the same argument as in section 2 that: 17

PROPOSITION 3.4. For F (x) a convex function on R+, i.e. F 00 (x)  0, Z

1 2AW

S

We can write

F (W ) ( + 00 )d  21 (F (1 ) + F (2 )): 1 L ? b 1 = 2LA + b; 2 = 2A W

and

W

?t2 = 2LA + u; ?t1 = 2AL ? u W

where

W

q

L2 ? 4AK AW : u= 2AW PROPOSITION 3.5. If K , W are symmetric, then 1  ?t2; i.e.

b  u: PROOF: By the argument of section 2, we may assume that the inscribed and circumscribed Wul shapes are centered at the origin, and thus ri ()  p()  re () for all , from which it follows that

?u ()  p() ? 2LA  u () W

for all . On I1,

W () ? a  0;

and thus which integrates to

? A1 W

?(p ? 2AL )(W ? a)  u (W ? a) W

Z

I1

(p ? 2LA )(W ? a)( + 00 )d  u1 ? 2a: W

On I2, we obtain similarly Z 1 (p ? 2LA )(W ? a)( + 00 )d  ?u2 + 2a: ?A W I2 W 18

Adding these, we get Z 1 ? A 1 (p ? 2AL )(W ? a)( + 00 )d  u(1 ? 2): W S W The right-hand side is 2ub, and the left-hand side simpli es to

L2 ? 4AK AW 2: = 2 u 2 2AW Thus

2u2  2ub; and hence u  b as desired. This proves the proposition. PROPOSITION 3.6. For convex K (symmetric or not) and W symmetric, then

1  ?t2; i.e.

b  u: PROOF: R We00symmetrize as in section 2, choosing points of @K corresponding to ,  +  with (p + p ) d equal to L 1 , L 2 respectively and with each half having areas A1 , A2 respectively. We choose  so that K1 , K2 with (2L 1 )2 ? 8A1AW = (2L 2 )2 ? 8A2 AW : We get two symmetric convex regions K1 , K2 with

L (Ki ) = 2L i and for i = 1; 2, and and

AKi = 2Ai L (K ) = L 1 + L 2

As in section 2, it is easy to see that

AK = A 1 + A 2 :

1(K )  21 (1(K1 ) + 2(K2 )): It follows from the symmetric case that p 1(K )  L2 A(K ) + ;

W

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where

2 2 = (2L 1 )4A?28A1 AW = (2L 2)4A?28A2 AW : W

W

We are thus reduced to showing that

( L

1 + L 2 )2 ? 4AW (A1 + A2 )  : 4A2W This in turn is equivalent to showing that 2L2 1 + 2L2 2 ? 4AW A1 ? 4AW A2  (L 1 + L 2 )2 ? 4AW A1 ? 4AW A2 ; which is equivalent to

(L 1 ? L 2)2  0: This completes the proof of the proposition. Department of Mathematics, UCLA, Los Angeles, CA 90095-1555 [email protected] [email protected] BIBLIOGRAPHY

[G] Gage, M. E., \An isoperimetric inequality with applications to curve shortening," Duke Math J. v. 50 (1983), 1225-1229. [O-M] Osher, S. and Merriman, B., \The Wul shape as the asymptotic limit of a growing crystalline interface," Asian J. of Math., v.1 (1997), 560-571. [S] Santalo, L., Integral Geometry and Geometric Probability, v.1 of Encyclopedia of Mathematics and its Applications, Addison Wesley, Reading, Mass. (1976). [Sc] Schneider, R., Convex bodies: the Brunn-Minkowski theory, Cambridge University Press, Cambridge, England (1993).

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