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January 3, 2017
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July 27, 2000
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December 31, 1967
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/V / /V V. 6 AUTHORS: Pestov,G., and i2RU.X--~ SOY/20-127-6-T/51 TITLE: On the Largest Possible Circle Imbedded in a Given Closed Curve PERIODICAL: Doklady Akademii nauk SSSR,1 t959,V01 127,11r 6,pp 1170-1172 (USSR) ABSTRACT: Theorem 1: Let the radius of curvature of a closed non inter- secting two times continuously differentiable curve r be,every- where not smaller than R ; then there always exists a circle 0 with the radius R lying completely in the closed domain bounded by el. 0 The curvature in the point P 61rie counted negatively if Irin the point P is convex with respect to the interior. Theorem 2t Let &,be closed, let it have no intersections, and let it be two tiines continuously differentiable. Let the curvature of r1be everywhere not larger than k. Then there exists a sphere of radius -1 lying completely in the interior of k the domain bounded by Card 1/2 'bn ihe Largest Possible Circle Imbedded in a SOV/20-127-6-7/51- Given Closed Curve Theorem 3: Under the assumptions of theorem 2 the length of is not smaller than !Nana the area bounded by Xis not smaller than Ic k k2 The author mentions A.I.Fet. PRESENTED: April 29, 1959, by S,L#Sobolev, Academician SUBMITTED: March 30, 1959 ~:~' ~ IONINIV, K. Some problems for convex surfaces with limitations for curvature. Sib.mat.zhur, 6 no.2;305-322 Mr-Ap 165. (MIRA 18:5) U4)- 16, S-'-/ 0 0 66441 AUTRORS t Uu4 K Suvorov, G.D. BOV/20-129-3-6;70 TITLBs On the Components of the Level Sets of the Function - Distance to a Plane Continuum PERIODICALt Doklady Akademii nauk SSSR, 1959,vol 129,Nr 3,pp 496 7498 (USSR) ABSTRACT: Let K be a bounded continuum in the plane P and r = ~(M,X)~ T(M) be the distance between the point Me P and K. The level set E r of 3(M) is the set of all M C-P for which t(M) r.- Theorem: Let EK be a component of E ; let G be that connected r oc r r PCV component of the open set F NE r which contains K; let G r be the t4 ot boundary of G. . Then all simple ends of G r contain one point each; it is E9~' G"and it holds: r r I. For all r> 0 the E d-may belong only to the following typ4st r 1. simple closed rectifiable Jordan curve; 2. simple open smooth Jordan are; 3. sum of finitely or countably many closed simple Jordan curves, smooth arcs, etc-; 4- point. II. The closed curves in the types 1 and 3 have no tangent in at most countably many points (corner points). The ramification Card 1/3 points in type 3 are points of regression. 66441 On the Components of the Level Sets of the SOV/20-129-3-6/70 Function-Distance to a Plane Continuum III. Components of the type 2 and 3 E~re possible at most for couzitably many level sets E If the co'mponenta of the.type.1 in a countable number appear af;o onlyfor countably many level sets, then at most countably many level sets have the components of the type 4- IV. Let A a A(o(,r, be the set of those points of 09~ the 04 r distance of which from E. is smaller than 04E .9 r. Let A be the boundary of A5 . Then A! \Er - r( cK, r, Sis a simple closed smooth Jordan curve. Let l(QqrqcS) be its length and 1(&