ON DISTANCE FUNCTIONS, AND ON POLAR SERIES OF CONVEX BODIES Submitted in partial fulfillment of the require ments for the degree of Doctor of Philosophy, in the Faculty of Pure Science, Columbia University, by Fritz Steinhardt ON DISTANCE FUNCTIONS. AND ON POLAR SERIES OF CONVEX BODIES 1. Introduction, Let F(Xj, ...,x ) be a real-valued continuous n function defined on the entire euclidean space E and satisfying (1.1) F(x) > 0 for (x) =£ (0), (1.2) F(A x) =• X F(x) for all real A» 0, where (x) Is short for (x , . .. ,x ) . A n The point-set S = {(*) 5 F(x) ^. lj in E i sa star body with respect to i ts interior point (0), and F(x) is the distance function (or gauge function) of S with respect to (0). The intersection of S with any line through (0) is a closed line segment. The convex kernel of 5 is the set of all Interior points of & which share with (0) the property just mentioned. If F(x) is also a convex function of (x): (1.3) F(x+ y) 2 S is introduced, and in terms of this concept, the effect of a translation of a star body on the associated distance function is examined. ( It will be recalled that for a convex body ^, and the associated support function H(x), the effect of a transla tion of 3Cby vector (b) is simply to replace H(x) by H(x) - ZLb. x..) Certain integral identities for distance functions are obtained as corollaries. Average radii and vectors of convex bodies are discussed briefly, and a new proof is given for a theo rem of Minkowski on projections of a convex body. The second part defines polar series of ster bodies (in ana logy to Brunn and Minkowski's linear series of convex bodies) and establishes their convexity properties. An application is made to a new characterization of the circle among convex curves, of the same type as Blaschke's characterization of ellipses in \j3~[> 2. Equivalent distance functions. Two distance functions F(x) and G(x) will be called equivalent if the two associated star bodies are congruent under a translation. Given a star body S with distance function F(x), let (-b) (-b, , .. . ,-b^ ) be an interior point of the convex kernel of , the family of equivalent distance functions 3 belonging to o • In the following, the distance functions will be assumed "sufficiently" differentiable, i.e., any derivatives to be written down are herewith assumed to exist. F^(x;b) will stand for ^F(x;b)/2> x., F^ (x;b) for ^F(x;b)/^b- , etc. +C If F(x) is the distance function of S and if the F. (x) exist (i = 1, ... ,v») , then (0) is interior to the convex kernel of S • In fact, the tangent hyperplane at any boundary point (x) of ) the ; boundary point of £ along the line of Ox, for any x <=/= 0. ft) -t {\>+A-\>) -1 Thus x^ =x.(F(x;b)) , and x " = x. (F(x;b+ii-b)) , since ,0) -U>) the boundary points x of 5 are characterized by F(x ;b)»l. Now x^ + c ' is approximately - i.e., up to terms of the second and higher orders in A^{ - the inter section z of Ox with the hyperpiane " S^ g&+M) T tangent to ,S L at x + A • b. Since T is parallel to the o>Q>) -(U hyperpiane tangent to o at x^ ' , it follows that T is normal to the vector (F (x;b) ,. . . ,F ( x;b) ), and is therefore given by ; ri T = {y; Z F,(x;b). [y -(x a,)+ ^ A b, )] = o] . y v L :i.es Thus if we set 2=/\x , then A satisfi vt ^.0) CM £L Fv(x;b).[ A x v ~"vxv ] = ^bcF-(x;b). V = l Now i x ,C W. F (x;b) = i . x l M. F (x(t) ;b).F(x41 ;b)=l (by v v v V=l V=l V V Euler' s theorem) , so that A = 1 4 A b. F. (x;b). r — It) Hence t = lit A b- F (x;b) 7 . x , and £ x (£+^^) .(F(x;b + ^ b))"1= ^ ( x j b ). (F(x;b +Ab))"1 == x c i = x4)[l44b P.(x;b)] + h.p.(^bj). { From this we obtain F(x;b-f^ b) - F(x;b)= -F(x;b) F. (x;b)[ H4 b F (x;b) c c c + h. p. (/\b )] . c 5 Division by A b; and passage to the limit as A b - —* 0 now yield t (3-D. If F(x;b) and G(x;b) agree at (x;0) for all (x), then so do all their x-uartials: 0 = F(x;0) - G( x; 0) = F. ( x; 0) - G-(x;0) = ... . By (3.1), then, setting H(x;b)= F(x;b) - G(x;b), we ob tain H (x;0)= F . (x;0) - G .. ( x. 0) = G( x; 0) G,(x;0) - F(x;0)* vvfC M+ £ rt+ F.(x;0)=» 0, and similarly for all higher b-partials at (x;0). This finishes the proof. We note that the homogeneity in (x) of F(x;b) is expressed by (3.2) F(x;b + Ax) = F(x;b) [ 1 + AF(x;b)J where the real number A is restricted to the neighborhood of 0 such that (b+Ax) lies within the convex kernel of S . Let G(x) be the r-th power (r^O) of a distance function, i. e., G(x) is any function which is positive for (x)^(0) and positive - homogeneous of degree r. Then for the equivalence family G(x;b) associated with G(x)=G(x;0), the relations cor responding to (3.1) and (3.2) are (3.1,r) G„ (x;b)= - G,A(x;b) G.(x;b), +C -K (3.2,r) G(x;b +-Xx) = G(x;b) [ 1 + AG ^ (x;b)] For use below, we obtain from (3.1,r) the further relations (the argument (x;b) is omitted) A >A + Gi ( 3.3, r) V-j " V" ~ ' ^ * I ™H °i 1' . . (2.-/1)1 As r i &„ • •- V*.G rGG- + 2G.G.1. 6 4. Some integral identities as corollaries. The volume of <5 is independent of (b). This fact, combined with (3»l) (or (3.1,r)) of Theorem 1, allows an immediate derivation of certain integral identities for homogeneous functions, avoiding cumbersome applications of Gauss' divergence theorem. Notation: vectors (\ ) — ( K , .. . , \ ) will be unit n vectors (I $1 = 1); duj = J.to(^ ) is the area element of the surface 5^. of the n-dimensional unit s-nere, Q =|( x) ; ^ x , ^= l} . With o> , the associated F(x;b), and G(x;b)= F*(x;b) as above, and V denoting volume, we have hence W ( 3 ^) /7> b; = 0, fv( S^^/^b- 2 b- -0 (etc.), combined with (3.1,r) and (3.3,r), yield (4.1) J (»-"-*)/*• ( ^ ) G- ( j ) olu^=0, i = l,...,n; G £ (4.2) | ci — ^ )A ) [ _ _ .j L o, G ( j (n+r 2)GiG rGC} / c = -n. i., j—x,..«,n. For greater formal generality, apoly (4.1) to G(x) — •k. tt (i - h) G (x). |x| (k any real number), wiich is itself > 0 for (x) ^ 0 and positive - homogeneous of degree r. This gives, setting t = 1 -f(n-l).k/r: (4.1,t) (1-t) / G ~t ( I ) G.( j )dw=(n-l + r - rt)„ i — t "si / ^Q ( 5 ) cLu> , i = 1, . . . , n and all real t, In (4.1,t), consider G(x) as a member of its equivalence family G(x;b) and differentiate with respect to b« , using (3.1,r) and 0 (3.3,r), t "> obtain - the argument ( j ) is omitted - (4.2,t) /G_trr(l-t) G.G.+GG.. -(n-2 +2r-tr) \ • GG-1OLJ = 0, i , jsl, ..., a and all real t. Finally, a ply (4.2) to G"(X) a,;d from the resulting identity subtract (4.2,t), to obtain (4.3, t) £a~t [(1-t) ^.G. + G( S-. -(n + r-rt) j - j. )]4)=^0, i t t i, j =1, ..., n and all real t. Further identities, involving higher partial derivatives, may be produced ?t will in the same .:.anner. A few special cases of the above identities are worth noting: 1. Apply (4.2,1) and 4.3,0) to a distance function F(x) (so that r =1), obtaining 1 (n-1)" J F..J^ ^ / ^ ^ ^ [ (n + Di.f.-LJF^; set I. = j and then sum over i '. 1 (n-lf / £ F. . ( j ) + rjG.G^ -rGG^.JoL). The identities whion arise from the invariance under translation of the surface S(g ) = / F "n { \ ) | V F (j ) ] °Lo , by the use of (3-1), are the same as those obtained by applying (4.1) to the distance function F(x). | F F ( X ) | ' ^ '^ . If ^ is convex, then log G(x;b) is. convex in (b) , for any r •* 0. (This follows easily from (3-1), (3.3). Note that for r = l, it is immediate also by geometric considerations.) 5. F(x;b), and projectijns of a convex body. If £> is a convex body, S ~ X > then (-b), in the associated family F(x;b), will range over the interior of 1C. For any (^ ), denote by TC,,, the projection of 1\. in direction ( j ); by <3" , the (n-l) - dimensional volume of *~y<~ .M . 77e shall derive a formula for ^c\ in terms of the ( F( ^ ;b). Fix ( \ ) (with \\\ = 1) . Any (b) in -% has a unique representation of the form b =» a- Aj, where ( - a ) is a boundary point of 7C , and A>0. Denote by dV_, a volume element of fii which contains the point ( - b ), and by dV_ ( ^ ), an ( ( n - 1 )- dimensional ) element of vol a ume of TO)which contains the projection of the boundary point ( - a ) of - 1. As k —* - 1 , the last integral becomes For ( - b) on the boundary^of ^C, the definition of F(x;b) may be extended in an obvious way, resulting in finite "boundary values" whenever the direction of (x) is sufficiently close to that of (b).

On Distance Functions, and on Polar Series of Convex Bodies

Similar On Distance Functions, and on Polar Series of Convex Bodies