SCIENTIFIC ABSTRACT IONIN, V. - IONKIN, A.YA

• Document Type: CREST
• Collection: Scientific Abstracts
• Document Number (FOIA) /ESDN (CREST): CIA-RDP86-00513R000518710014-4
• Release Decision: RIF
• Original Classification: S
• Document Page Count: 100
• Document Creation Date: January 3, 2017
• Document Release Date: July 27, 2000
• Sequence Number: 14
• Publication Date: December 31, 1967
• Content Type: SCIENTIFIC ABSTRACT

/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(&