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" 20 for some AE R. Are the converses of these statements true? a 11
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" 20 for some AE R. Are the converses of these statements true? a 11
Advanced Engineering Mathematics
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ISBN:9780470458365
Author:Erwin Kreyszig
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Chapter2: Second-order Linear Odes
Section: Chapter Questions
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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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