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Hyperbolic sets. Show that the hyperbolic set {r € R² | x1x2 > 1} is convex. As a generalization, show that {x € R | II", Ti > 1} is convex. Hint. If a, b > 0 and 0 <0<1, then aºb'-º < 0a + (1 – 0)b; see §3.1.9.

Hyperbolic sets. Show that the hyperbolic set {r € R² | x1x2 > 1} is convex. As a generalization, show that {x € R | II", Ti > 1} is convex. Hint. If a, b > 0 and 0 <0<1, then aºb'-º < 0a + (1 – 0)b; see §3.1.9.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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2.10 Solution set of a quadratic inequality. Let C C R" be the solution set of a quadratic inequality, C = {x € R" | a" Ax + b" x + c < 0}, with A E S", b E R", and c E R. (a) Show that C is convex if A E 0. (b) Show that the intersection of C and the hyperplane defined by g"x +h = 0 (where g + 0) is convex if A + Agg" > 0 for some AE R. Are the converses of these statements true? 2.11 Hyperbolic sets. Show that the hyperbolic set {x E R² | x1x2 > 1} is convex. As a generalization, show that {x E R | II", xi > 1} is convex. Hint. If a, b > 0 and 0 <0< 1, then aºb'=° < 0a+ (1 – 0)b; see §3.1.9. i=1 2.12 Which of the following sets are convex? (a) A slab, i.e., a set of the form {x € R" | a < ax < }. (b) A rectangle, i.e., a set of the form {x € R" | ai <xi < Bi, i = 1, ...,n}. A rectangle is sometimes called a hyperrectangle when n > 2. (c) A wedge, i.e., {r € R" | afx < b1, afa < b2}. (d) The set of points closer to a given point than a given set, i.e., {x| ||x – xo||2 < |r – y||2 for all y E S} where S C R". (e) The set of points closer to one set than another, i.e., {x | dist(x, S) < dist(x,T)}, where S, T C R", and dist(r, S) = inf{||r – 2||2 | 2 E S}.
Transcribed Image Text:2.10 Solution set of a quadratic inequality. Let C C R" be the solution set of a quadratic inequality, C = {x € R" | a" Ax + b" x + c < 0}, with A E S", b E R", and c E R. (a) Show that C is convex if A E 0. (b) Show that the intersection of C and the hyperplane defined by g"x +h = 0 (where g + 0) is convex if A + Agg" > 0 for some AE R. Are the converses of these statements true? 2.11 Hyperbolic sets. Show that the hyperbolic set {x E R² | x1x2 > 1} is convex. As a generalization, show that {x E R | II", xi > 1} is convex. Hint. If a, b > 0 and 0 <0< 1, then aºb'=° < 0a+ (1 – 0)b; see §3.1.9. i=1 2.12 Which of the following sets are convex? (a) A slab, i.e., a set of the form {x € R" | a < ax < }. (b) A rectangle, i.e., a set of the form {x € R" | ai <xi < Bi, i = 1, ...,n}. A rectangle is sometimes called a hyperrectangle when n > 2. (c) A wedge, i.e., {r € R" | afx < b1, afa < b2}. (d) The set of points closer to a given point than a given set, i.e., {x| ||x – xo||2 < |r – y||2 for all y E S} where S C R". (e) The set of points closer to one set than another, i.e., {x | dist(x, S) < dist(x,T)}, where S, T C R", and dist(r, S) = inf{||r – 2||2 | 2 E S}.
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