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chore(analysis/convex): trivial generalizations of ℝ (#9298)
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/- | ||
Copyright (c) 2019 Yury Kudriashov. All rights reserved. | ||
Released under Apache 2.0 license as described in the file LICENSE. | ||
Authors: Yury Kudriashov, Yaël Dillies | ||
-/ | ||
import analysis.convex.basic | ||
import data.complex.module | ||
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/-! | ||
# Convexity of half spaces in ℂ | ||
The open and closed half-spaces in ℂ given by an inequality on either the real or imaginary part | ||
are all convex over ℝ. | ||
-/ | ||
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lemma convex_halfspace_re_lt (r : ℝ) : convex ℝ {c : ℂ | c.re < r} := | ||
convex_halfspace_lt (is_linear_map.mk complex.add_re complex.smul_re) _ | ||
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lemma convex_halfspace_re_le (r : ℝ) : convex ℝ {c : ℂ | c.re ≤ r} := | ||
convex_halfspace_le (is_linear_map.mk complex.add_re complex.smul_re) _ | ||
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lemma convex_halfspace_re_gt (r : ℝ) : convex ℝ {c : ℂ | r < c.re } := | ||
convex_halfspace_gt (is_linear_map.mk complex.add_re complex.smul_re) _ | ||
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lemma convex_halfspace_re_ge (r : ℝ) : convex ℝ {c : ℂ | r ≤ c.re} := | ||
convex_halfspace_ge (is_linear_map.mk complex.add_re complex.smul_re) _ | ||
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lemma convex_halfspace_im_lt (r : ℝ) : convex ℝ {c : ℂ | c.im < r} := | ||
convex_halfspace_lt (is_linear_map.mk complex.add_im complex.smul_im) _ | ||
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lemma convex_halfspace_im_le (r : ℝ) : convex ℝ {c : ℂ | c.im ≤ r} := | ||
convex_halfspace_le (is_linear_map.mk complex.add_im complex.smul_im) _ | ||
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lemma convex_halfspace_im_gt (r : ℝ) : convex ℝ {c : ℂ | r < c.im} := | ||
convex_halfspace_gt (is_linear_map.mk complex.add_im complex.smul_im) _ | ||
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lemma convex_halfspace_im_ge (r : ℝ) : convex ℝ {c : ℂ | r ≤ c.im} := | ||
convex_halfspace_ge (is_linear_map.mk complex.add_im complex.smul_im) _ |
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