TY - JOUR

T1 - The equal tangents property

AU - Jerónimo-Castro, Jesús

AU - Ruiz-Hernández, Gabriel

AU - Tabachnikov, Sergei

N1 - Funding Information:
Consider the family of “circles", tangent to ℓ at point p. In the complement of p, these curves form a smooth foliation F. Since the two tangent segments to γ from every point x ∈ ℓ have equal lengths, the curve γ is everywhere tangent to the leaves of the foliation F. It follows that γ coincides with a leaf, that is, γ is a “circle". Since γ is a closed curve, it is indeed a circle. ✷ Acknowledgments. The first author was partially supported by CONACYT SNI 38848 and the third author was partially supported by the Simons Foundation grant No 209361 and by the NSF grant DMS-1105442.

PY - 2014/7/1

Y1 - 2014/7/1

N2 - Let M be a C2-smooth strictly convex closed surface in ℝ3 and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface containing M or a plane, then M is a Euclidean sphere. Moreover, we shall see that the situation in the Euclidean plane is very different.

AB - Let M be a C2-smooth strictly convex closed surface in ℝ3 and denote by H the set of points x in the exterior of M such that all the tangent segments from x to M have equal lengths. In this note we prove that if H is either a closed surface containing M or a plane, then M is a Euclidean sphere. Moreover, we shall see that the situation in the Euclidean plane is very different.

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U2 - 10.1515/advgeom-2013-0011

DO - 10.1515/advgeom-2013-0011

M3 - Article

AN - SCOPUS:84925373760

VL - 14

SP - 447

EP - 453

JO - Advances in Geometry

JF - Advances in Geometry

SN - 1615-715X

IS - 3

ER -