# Get A. D. Alexandrovs length manifolds with one-sided bounded PDF

By Berestovskii V. N.

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Extra resources for A. D. Alexandrovs length manifolds with one-sided bounded curvature

Example text

143) are specified by those values of t which satisfy the equation (d + t b) · ˜ · (d + t b) = (b · ˜ · b)t 2 + 2(b · ˜ · d)t + (d · ˜ · d) = 1. 141) at two distinct points. 141), and if it has no real roots then the line does not meet the ellipsoid. 145) is zero. 145) is zero as well, because, in this case, the root has to have double multiplicity, that is, b · ˜ · d = 0. 146), provides the following equation of the tangent line at the point d: r · ˜ · d = 1. 141) at the point d. 150) which assigns the unit normal nˆ to every point d on the surface of the ellipsoid.

165) and similarly we can show that s · t = ε2 . 12. 4 The plane that is perpendicular to the direction of any position vector of an ellipsoid and passes through the Kelvin image of the position vector, with respect to a sphere of radius ε, is tangent to the reciprocal ellipsoid. 4 implies that, if we take the Kelvin image [337–339] of an ellipsoid with respect to a co-centered sphere of radius ε, and consider the family of the planes that are perpendicular to the directions of the position vectors and pass through the Kelvin images of these position points, then the envelope of this family of planes generates the reciprocal ellipsoid.

164) shows that the normal vector to the tangent plane at t is t, which completes the proof. 12 Reciprocity for the directions of tangency and support. 165) and similarly we can show that s · t = ε2 . 12. 4 The plane that is perpendicular to the direction of any position vector of an ellipsoid and passes through the Kelvin image of the position vector, with respect to a sphere of radius ε, is tangent to the reciprocal ellipsoid. 4 implies that, if we take the Kelvin image [337–339] of an ellipsoid with respect to a co-centered sphere of radius ε, and consider the family of the planes that are perpendicular to the directions of the position vectors and pass through the Kelvin images of these position points, then the envelope of this family of planes generates the reciprocal ellipsoid.