/F2 1 Tf 20.6626 0 0 20.6626 278.838 258.078 Tm (i)Tj 1 i ($$)Tj The subspace Y is a convex set if for each pair of points a, b in Y such that a ≤ b, the interval [a, b] = {x ∈ X | a ≤ x ≤ b} is contained in Y. A vector x0 is an interior point of the set X, if there is a ball B(x0,r) contained entirely in the set X Def. 0 Tc 1.8059 0 TD /F2 1 Tf [(CHAPTER)-327.3(3. -13.8787 -1.2052 TD (H)Tj (m)Tj /F4 1 Tf /F2 1 Tf 0 Tc /F2 1 Tf 0 Tc /F2 1 Tf 414.25 597.477 l )Tj 1.0789 0 TD If a and b are points in a vector space the points on the straight line between a and … 4.1503 0 TD -17.8918 -2.4431 TD {\displaystyle \operatorname {rec} A\cap \operatorname {rec} B} 0 -1.2057 TD 0.6608 0 TD 1.6896 0 TD /F2 1 Tf /F2 5 0 R 1.0689 0 TD /F3 1 Tf -0.0001 Tc (where)Tj 112.707 625.823 m [(. (i)Tj (i)Tj /F3 1 Tf /F2 1 Tf 329.211 597.477 m (102)Tj /F1 4 0 R (H)Tj [(is)-306.4(e)50.2(q)0.1(ual)]TJ /F3 1 Tf /F3 1 Tf 0.5314 0 TD 0.389 0 TD 1.0763 0 TD [(ve)50.1(ctors)-306.9(i)-0.1(n)]TJ 15.0861 0 TD /F1 4 0 R [(b)50.2(e)-386.6(a)-386.3(family)-386(o)0(f)-386.4(p)50.1(oints)-386.6(i)0(n)]TJ 1.0528 0 TD [(can)-377.2(b)-26.1(e)-377.6(w)-0.1(ritten)-377.2(as)-377.1(a)-377.2(c)0.1(on)26.1(v)26.2(e)0.1(x)-377.2(c)0.1(om-)]TJ Convexity can be extended for a totally ordered set X endowed with the order topology.[19]. (i)Tj -7.9956 -2.363 TD [(and)-301.8(is)-301.8(denoted)-301.8(b)26.1(y)]TJ 1.0554 0 TD (L,)Tj )]TJ (f)Tj 0.585 0 TD -0.0003 Tc (m)Tj (V)Tj [(c)50.1(onvex)-420.3(hul)-50.1(l)-420.4(of)]TJ 0 Tc 1.525 0 TD -19.3219 -1.2052 TD /F4 1 Tf (\()Tj R 379.485 628.847 m ()Tj 0 g [($$,)-285.2(w)26(e)-280.5(can)-280.6(d)-0.1(e“ne)-280.5(the)-280.5(t)26.1(w)26(o)]TJ 0 g 0 g /F4 1 Tf /F8 1 Tf /F8 16 0 R Tools: De nitions ofconvex sets and functions, classic examples 24 2 Convex sets Figure 2.2 Some simple convex and nonconvex sets. [(0,)-423.2(then)]TJ [(of)-359.4(dimen-)]TJ (J)Tj /F5 1 Tf (i)Tj /F7 1 Tf /F7 1 Tf /F10 24 0 R 1.0554 0 TD 0.8768 0 TD [(con)26.1(v)-13($$)]TJ (f)Tj /F4 1 Tf 15.1802 0 TD 0.5893 0 TD 0.6608 0 TD ( Convex Sets and Convex Functions 1 Convex Sets, In this section, we introduce one of the most important ideas in economic modelling, in the theory of optimization and, indeed in much of modern analysis and computatyional mathematics: that of a convex set. 8.0035 0 TD (. 0.5798 0 TD 8.4369 0 TD 14.3462 0 0 14.3462 325.017 573.402 Tm 1.1068 0 TD 0.7936 0 TD -5.5701 -2.8447 TD 1.9728 0 TD (H)Tj X [(1o)393.7(ft)393.8(h)393.7(e)]TJ 0.0001 Tc (\()Tj (\()Tj /ProcSet [/PDF /Text ] 0.0001 Tc 0 Tc /F4 1 Tf = [(Car)50.1(a)-0.1(th)24.8(´)]TJ /F2 1 Tf (=)Tj 14.3462 0 0 14.3462 155.538 573.402 Tm [(,o)261.6(f)]TJ /F5 1 Tf {\displaystyle s_{0}\in S} )Tj (+)Tj 0.8128 0 TD S (m)Tj 20.6626 0 0 20.6626 72 702.183 Tm 0.6943 0 TD [(i,)-166.5(j)]TJ /F4 1 Tf )]TJ ET 354.609 710.863 329.211 685.464 329.211 654.17 c 20.6626 0 0 20.6626 208.116 406.2631 Tm -22.3918 -2.3625 TD A convex set is not connected in general: a counter-example is given by the space Q, which is both convex and totally disconnected. f f /F3 1 Tf 39 0 obj 1.7118 0 TD BT /F2 1 Tf 14.3462 0 0 14.3462 170.163 330.0511 Tm (. (I)Tj /GS1 gs 357.557 625.823 l /F5 1 Tf /F8 16 0 R 0.8537 0 TD 0.2775 Tc endobj 20.6626 0 0 20.6626 249.741 576.498 Tm /F4 1 Tf /F5 1 Tf /F4 1 Tf ()Tj 0.6669 0 TD 0.8886 0 TD ∈ [($$)-350(i)0(s)-350(t)0.2(he)-349.6(c)50.2(onvex)-350.1(hul)-50(l)-350.1(of)]TJ ET 0.0001 Tc 0.4504 Tc [(asserts)-461.7(that)-462.1(it)-462.1(is)-461.7(enough)-461.8(to)-462.2(consider)-461.7(con)26(v)26.1(ex)]TJ 0.333 Tc 11.9551 0 0 11.9551 72 736.329 Tm -21.8495 -1.2057 TD 0.7183 0 TD ()Tj 0 G /F2 1 Tf 0 -1.2052 TD 49, 2003 Support functions of general convex sets 307 denote the algebra structure on R given by the join semilattice operation x+y = max{x,y} and thebinary operations p of (2.3) forp in I .ThenD is a modal. 8.1141 0 TD -22.2956 -1.2052 TD 0.6669 0 TD 0 -1.2052 TD 0.5893 0 TD ()Tj [(v)26.1(e)0(x)-305.4(s)0(ubset,)]TJ /F7 1 Tf 0 J 0 j 0.996 w 10 M []0 d 0.585 0 TD 0.9975 0 TD /F2 5 0 R 6.6161 0 TD (V)Tj 14.3462 0 0 14.3462 448.218 372.144 Tm 20.6626 0 0 20.6626 169.488 551.595 Tm 0 Tc [(Given)-465.3(an)-464.9(ane)-465.2(sp)50(ac)50.1(e)]TJ (E)Tj 357.557 597.477 l -10.1165 -1.2057 TD 0.6608 0 TD 0.5763 0 TD 0.3338 0 TD /F4 1 Tf 0.876 0 TD 20.6626 0 0 20.6626 333.243 652.368 Tm 2.8875 0 TD /F4 1 Tf rec ()Tj 0.3338 0 TD /F3 1 Tf /F4 1 Tf 0.3999 0 TD (a)Tj [(com)26(b)0(inations)-301.3(of)]TJ 20.6626 0 0 20.6626 124.938 436.3051 Tm (j)Tj 0.49 0 TD 20.6626 0 0 20.6626 379.566 407.8741 Tm (\))Tj /F4 1 Tf )Tj /F8 1 Tf /F4 1 Tf (i)Tj 20.6626 0 0 20.6626 394.875 267.4921 Tm 0.0001 Tc -14.8207 -2.8447 TD (i)Tj [(Theorem)-375.9(3.2.6)]TJ 0.889 0 TD (S)Tj Rawlins G.J.E. [(denoted)-446.1(b)26(y)]TJ /F3 1 Tf (S)Tj rec 20.6626 0 0 20.6626 195.444 292.4041 Tm 2 2 /F3 1 Tf ≤ /F2 1 Tf /F7 1 Tf [(,)-315.4(t)0.2(he)-306.5(c)50.2(one,)]TJ (E)Tj {\displaystyle R\leq {\frac {\sqrt {3}}{3}}D}, r /F5 1 Tf (>)Tj BT (Š)Tj BT /F2 1 Tf /F5 1 Tf 14.3462 0 0 14.3462 438.561 341.274 Tm /F5 1 Tf /F5 1 Tf [(,)-322.2(t)0.1(hat)-318.3(is,)-322.6(linear)-318.3(c)0.1(om)26(binations)-317.9(of)-318.3(the)]TJ (f)Tj •Example: subset sum problem •Given a set of integers, ... •Convex functions can’t approximate non-convex ones well. 0.389 0 TD (I)Tj (=1)Tj 0.6991 0 TD 0.6608 0 TD /F4 1 Tf 0 Tc 0.0001 Tc 0.0001 Tc /F5 1 Tf 6 LECTURE 1. (for)Tj /F4 1 Tf [(+)-222.3(1)-301.9(p)-26.2(o)-0.1(in)26(ts. 430.492 612.855 429.555 613.792 428.4 613.792 c 20.6626 0 0 20.6626 355.869 663.519 Tm /F1 1 Tf 3.175 0 TD [(a,)-166.6(b)]TJ 0 Tc (I)Tj /F4 1 Tf (f)Tj /F4 1 Tf 11.7569 0 TD /F4 1 Tf 2.025 0 TD /F4 1 Tf /F1 4 0 R (\). /F5 1 Tf endobj (ar´)Tj 0 -1.2052 TD An example of a recent result in this more general setting is the following theorem by Novick: Given 7.2k pairwise disjoint convex sets in the plane there is a set in the family that is disjoint to the convex hull of k other sets in the family. 0.0041 Tc 0 Tc ()Tj 14.3462 0 0 14.3462 194.139 660.4141 Tm >> 17.5298 0 TD 0.0001 Tc And equivalently if f(x) is quasi-convex, -f(x) is quasi-concave. {\displaystyle r+R\leq D}, D D 0.7597 0 TD (i)Tj (E)Tj 0.0001 Tc 0.0001 Tc /F2 1 Tf A polygon that is not a convex polygon is sometimes called a concave polygon,[3] and some sources more generally use the term concave set to mean a non-convex set,[4] but most authorities prohibit this usage. 0.389 0 TD 1.2209 0 TD -0.0001 Tc -21.4158 -1.2052 TD /F8 16 0 R /Font << 0 Tc 1.2113 0.95 TD 0.0001 Tc /F3 1 Tf The notion of a convex set can be generalized as described below. ()Tj )Tj /F4 1 Tf ()Tj (If)Tj Minkowski addition behaves well with respect to the operation of taking convex hulls, as shown by the following proposition: Let S1, S2 be subsets of a real vector-space, the convex hull of their Minkowski sum is the Minkowski sum of their convex hulls. 14.3462 0 0 14.3462 458.802 515.6041 Tm 2 0.3338 0 TD /F2 1 Tf 0.0001 Tc )Tj 1.1604 0 TD 19 0 obj 19.3423 0 TD /F2 1 Tf /F3 1 Tf 387.657 628.847 l 0.3338 0 TD 0.9361 0 TD for all z with kz − xk < r, we have z ∈ X Def. (subsets)Tj /F4 1 Tf 0 Tc X /F6 9 0 R BT 0.5001 0 TD 14.269 0 TD << 20.6626 0 0 20.6626 177.273 333.1561 Tm 0.5001 0 TD /F4 1 Tf /F4 1 Tf 0 Tc 0 Tc /F6 9 0 R 0.0002 Tc 0.2777 Tc (103)Tj 0 Tc 14.3462 0 0 14.3462 281.808 240.78 Tm (E)Tj 0 Tc 0.6608 0 TD (,...,S)Tj /F4 1 Tf /F2 1 Tf 0 Tc /F2 1 Tf /ProcSet [/PDF /Text ] ($$)Tj 1 convex sets. 0 Tc -22.3781 -1.7841 TD 0.9448 0 TD 0.8341 0 TD >> Simple examples of convex sets are: The empty set ;, the singleton set fx 0g, and the complete space Rn; Lines faT x= bg, line segments, hyperplanes fAT x= bg, and halfspaces fAT x bg; Euclidian balls B(x 0; ) = fxjjjx x 0jj 2 g. 14 0 obj )]TJ /F4 1 Tf A set C is strictly convex if every point on the line segment connecting x and y other than the endpoints is inside the interior of C. A set C is absolutely convex if it is convex and balanced. (. 1.1691 0 TD 16.6059 0 TD 2.2015 0 TD 0.4587 0 TD -0.0001 Tc ()Tj 0.3541 0 TD /F7 1 Tf /F2 1 Tf (S)Tj /F5 1 Tf /F9 20 0 R (1)Tj )Tj (i)Tj 0.0001 Tc 14.3462 0 0 14.3462 196.461 375.2761 Tm Equivalently, a convex set or a convex region is a subset that intersect every line into a single line segment (possibly empty). /F9 1 Tf [(m)25.9(u)-0.1(st)-391.6(b)-26.2(e)-392(anely)]TJ /F2 1 Tf ()Tj -0.0003 Tc 0.4587 0 TD 0 Tc [(eo)50.1(dory�s)-350(T)0.1(he)50.2(or)50.2(em)]TJ 14.3462 0 0 14.3462 219.006 573.402 Tm /F7 10 0 R 0.6773 0 TD /F2 1 Tf [(is)-306.8(a)-307(c)50.2(onvex)-306.9(c)50.2(o)0(mbina-)]TJ -0.0002 Tc /Length 5100 427.245 613.792 426.308 612.855 426.308 611.7 c -19.6267 -1.2052 TD (f)Tj 0.3337 0 TD 20.6626 0 0 20.6626 527.418 455.106 Tm ()Tj (i)Tj 220.959 648.5 m 0.0001 Tc >> 1.4562 0 TD 0 Tc -0.1302 -0.2529 TD 0 g /F2 1 Tf [(,)-349.8(and)]TJ 1.3559 0 TD -8.4369 -1.2052 TD )Tj 1.0554 0 TD [(it)-310(is)-310.5(enough)-309.7(to)-310.1(assume)-310.1(that)]TJ (. /F2 5 0 R /F3 1 Tf ($$)Tj 11.9551 0 0 11.9551 72 736.329 Tm (=)Tj (\)=)Tj S /F4 1 Tf (E)Tj 0.7836 0 TD 0 Tc [(c)50.2(onvex)-390.6(c)50.2(one)]TJ (If)Tj 0.3809 0 TD S 1.4958 0 TD 0.6608 0 TD /F5 1 Tf /F5 1 Tf 27 0 obj (\))Tj /F2 1 Tf endobj 3.3536 0 TD /F5 1 Tf [(family)-342.4(of)-342.8(half-spaces)-342.4(asso)-26.2(ciated)-342.4(with)-342.4(h)26(y)-0.1(p)-26.2(erplanes)-342.4(p)-0.1(la)26.1(y)-342.4(a)]TJ 0 Tc 14.3462 0 0 14.3462 244.179 538.1671 Tm /F2 1 Tf 3.5383 0 TD K >> 8.6743 0 TD -18.5359 -1.2052 TD 0 /F2 1 Tf /F4 1 Tf >> (S)Tj (and)Tj /F5 1 Tf 0.8564 0 TD [(3.2)-1125.1(C)0.1(arath)24.3(«)]TJ 0.8716 0 TD /F4 1 Tf (I)Tj Some other properties of convex sets are valid as well. 58 LECTURE 3. /F2 1 Tf ()Tj 0.1666 Tc -21.7937 -1.2057 TD (c)Tj /F5 1 Tf [(set)-301.9(of)-301.8(all)-301.8(p)-26.2(ositiv)26.1(e)-302.3(linear)-301.9(c)0(om)25.9(binations)-301.4(of)-301.8(v)26.1(ectors)-301.9(in)]TJ BT (S)Tj (104)Tj /F4 1 Tf (Š)Tj /F4 1 Tf /F5 1 Tf 0.7814 0 TD /F3 1 Tf 14.3462 0 0 14.3462 134.721 433.2001 Tm /F2 1 Tf 0 g Lecture 2 Open Set and Interior Let X ⊆ Rn be a nonempty set Def. 14.3462 0 0 14.3462 202.761 289.299 Tm 0 Tc 0.8563 0 TD /F9 20 0 R /F3 1 Tf /F5 1 Tf /F4 1 Tf /ProcSet [/PDF /Text ] 1.494 w 6.7293 0 TD /F4 1 Tf )-558.9(T)0.1(he)-386.6(family)]TJ 7.6254 0 TD 20.6626 0 0 20.6626 355.896 383.277 Tm /F8 1 Tf 20.6626 0 0 20.6626 182.34 663.519 Tm [(and)-420.4(th)26(us,)-449.9(a)-420.8(set)]TJ -0.0001 Tc (\))Tj /F7 1 Tf [(Lemma)-375.4(3.1.2)]TJ /F1 1 Tf [(its)-301.9(extremal)-301.8(p)-26.2(o)-0.1(in)26(ts)-301.9($$see)-301.9(Berger)-301.9([)]TJ 14.3462 0 0 14.3462 225.432 548.499 Tm 20.6626 0 0 20.6626 232.173 292.4041 Tm 1.1604 0 TD << /F2 1 Tf 0 -1.8712 TD 0.6991 0 TD 4 /F2 1 Tf /F3 1 Tf (+)Tj 0 g /F4 1 Tf 45 0 obj /F2 1 Tf /F1 4 0 R 1.0014 -1.7841 TD ()Tj 14.3462 0 0 14.3462 109.458 587.3701 Tm /F4 1 Tf (I)Tj 0.0001 Tc )Tj 0 Tc 20.6626 0 0 20.6626 72 702.183 Tm C. ... all level sets are compact. 4.7292 0 TD /F4 1 Tf /F2 1 Tf We can inscribe a rectangle r in C such that a homothetic copy R of r is circumscribed about C. The positive homothety ratio is at most 2 and:[10], The set /F5 1 Tf /F4 1 Tf [(line)50.2(ar)-365.8(c)50.2(o)0(mbinations)]TJ [(CHAPTER)-327.3(3. 0 -1.2052 TD 1.369 0 TD ()Tj ()Tj 6.4362 0 TD /F5 1 Tf 0.0001 Tc (de“ning)Tj )Tj 14.3462 0 0 14.3462 478.044 674.175 Tm [(EODOR)81.5(Y)0(’S)-326.3(THEOREM)]TJ (L)Tj 1.0903 0 TD /F4 1 Tf /F6 1 Tf ()Tj /F4 1 Tf (m)Tj 0.9875 0 TD /F4 1 Tf /F2 1 Tf /F4 7 0 R 0.0001 Tc 6.6699 0.2529 TD 0.2503 Tc [(Figure)-325.9(3.2:)-436.4(The)-325.9(t)27(w)27.4(o)-326.5(half-spaces)-326.7(determined)-325.5(b)26.8(y)-326.4(a)-326.5(h)26.8(y)0.4(p)-27.4(e)0.1(rplane,)]TJ (+)Tj 0 Tc /F5 1 Tf (S)Tj /F4 7 0 R 0 Tc [14][15], The Minkowski sum of two compact convex sets is compact. (=)Tj 0.7087 0 TD 4.7126 0 TD ⁡ /F4 1 Tf /Font << [(,)-427.2(t)0(here)-402.1(is)-402.5(a)]TJ 391.038 591.807 l >> (H)Tj /F4 1 Tf (S)Tj 0.862 0 TD -20.6884 -1.2052 TD (:)Tj /F2 1 Tf /F2 1 Tf 0.6669 0 TD 4.8001 0 TD 5.5036 0 TD − /F2 1 Tf -19.3257 -1.2052 TD /F4 1 Tf /F4 1 Tf ($$)Tj /F2 1 Tf S 20.6626 0 0 20.6626 140.004 436.3051 Tm [(,o)273(r)]TJ /F4 1 Tf 14.3462 0 0 14.3462 303.831 638.9041 Tm (C)Tj /F4 1 Tf /F2 1 Tf D -6.7764 -2.3625 TD 0.9822 0 TD ET − ()Tj -0.0002 Tc (i)Tj /F2 1 Tf (,)Tj R [(in)26(tersection)-332.9(o)-0.1(f)-333.2(a)-0.1(ll)-333.2(con)26(v)26.1(ex)-332.8(sets)-332.8(con)26(t)0(aining)]TJ [(that)-224.8(the)-224.4(a)-0.1(ne)-224.8(s)0(pace)]TJ 0.7836 0 TD -18.5371 -1.2052 TD In geometry, a subset of a Euclidean space, or more generally an affine space over the reals, is convex if, given any two points, it contains the whole line segment that joins them. /Font << /ExtGState << 20.6626 0 0 20.6626 421.299 541.272 Tm 2 . (+1)Tj Notice that while deﬁning a convex set, /Length 4531 /F5 1 Tf [(This)-339.3(theorem)-339.5(due)-339.8(to)-339.4(B´)]TJ -17.5776 -1.2052 TD [(Car)50.1(a)-0.1(th)24.8(´)]TJ 20.6626 0 0 20.6626 417.555 258.078 Tm )Tj (S)Tj /F4 1 Tf Then, given any (nonempty) subset S of E, there is a smallest convex set containing S denoted by C(S)(or conv(S)) and called the convex hull of S (namely, theintersection of all convex sets containing S).The aﬃne hull of a subset, S,ofE is the smallest aﬃne set contain- 0.0001 Tc 0.5554 0 TD 0 Tc Convex Optimization - Polyhedral Set - A set in $\mathbb{R}^n$ is said to be polyhedral if it is the intersection of a finite number of closed half spaces, i.e., /F2 1 Tf (S)Tj (|)Tj (+1)Tj 1.0559 0 TD 0.0001 Tc 20.6626 0 0 20.6626 347.589 529.6981 Tm (Š)Tj 0 Tc endstream stream Suppose there is a smaller convex set S. 20.6626 0 0 20.6626 533.799 199.0651 Tm 329.211 597.477 m 0 Tc 0 Tc 0 0 1 rg 14.3462 0 0 14.3462 358.362 404.769 Tm 20.6626 0 0 20.6626 221.58 663.519 Tm /F3 1 Tf ) 0 Tc 4.7701 0 TD /GS1 11 0 R (with)Tj (i)Tj Given a set X, a convexity over X is a collection of subsets of X satisfying the following axioms:[8][9][20]. /F6 1 Tf Let S be a vector space or an affine space over the real numbers, or, more generally, over some ordered field. /GS1 gs /F5 1 Tf 20.6626 0 0 20.6626 323.226 195.0601 Tm 20.6626 0 0 20.6626 407.628 344.3701 Tm (m)Tj (E,)Tj -0.0003 Tc It is the smallest convex set containing A. [(\),)-273.1(is)-266.6(the)]TJ (E)Tj 0.6669 0 TD (f)Tj 0.6904 0 TD 0.3541 0 TD 0.3338 0 TD (is)Tj (v)Tj 2.8204 0 TD << /F4 1 Tf 0.9722 -1.7101 TD [(has)-224.2(dimension)]TJ 0 -1.2052 TD /F4 1 Tf 0.9443 0 TD 0.3541 0 TD ⊆ 0 g 0.0001 Tc (H)Tj 13.4618 0 TD (E)Tj 0 Tw 354.609 710.863 329.211 685.464 329.211 654.17 c /F2 1 Tf )-467.2(When)]TJ [(eo)50.1(dory�s)-249.8(T)0.1(he)50.2(or)50.2(em,)-270.1(R)50.1(adon)100.1(�s)-249.8(The-)]TJ 0.0001 Tc ([)Tj (i)Tj (. /F2 1 Tf 0 Tc 0.2777 Tc -0.0001 Tc 0 g (S,)Tj The Kepler-Poinsot polyhedra are examples of non-convex sets. [(h)26.1(y)0(p)-26.1(erplane)]TJ [(Helly�s)-372(theorem)-371.8(kno)26.1(wn)-371.6(as)-372(Tv)26.1(erb)-26.2(erg�s)-372(theorem)-371.8($$see)-372(Sec-)]TJ ()Tj /F4 1 Tf 0 Tc -0.0003 Tc 1.6025 0 TD [(Note)-321.8(t)0.1(hat)-322.2(a)-321.8(c)0.1(one)-322.3(alw)26.1(a)26.2(ys)-321.8(con)26.1(t)0.1(ains)-321.8(0. 4.4878 0 TD -21.7439 -2.5664 TD /F5 1 Tf >> [(pro)26.1(v)26.1(e)-359.8(it)-360.2(here. (f)Tj 20.6626 0 0 20.6626 199.431 541.272 Tm ()Tj /F4 1 Tf 0.6904 0 TD 414.25 597.477 l A set C Rnis convex if 8x1;x2 2C;8 2[0;1] we have that x = x1 +(1 )x2 2C: Intuitively, a set is convex if the line segment between any two of its points is in the set. )Tj /F4 1 Tf 7.3348 0 TD /F11 1 Tf 0.2226 Tc -22.3781 -1.7837 TD /F2 1 Tf [(a,)-166.6(b)]TJ /F4 1 Tf 10.0333 0 TD 0.3541 0 TD (,...,a)Tj 0.5893 0 TD 0.0001 Tc >> 0.6608 0 TD 11.9551 0 0 11.9551 72 736.329 Tm endstream ()Tj /F7 10 0 R ()Tj 0.3499 Tc /F2 1 Tf 0.6608 0 TD (\()Tj /F2 1 Tf (i)Tj 14.7128 0 TD [(the)-301.4(union)-301.9(of)-301.4(tetrahedra)-301.5(\(including)-301.9(in)26(terior)-301.4(p)-26.2(oin)26(ts$$)-301.9(whose)]TJ [(\). 1.0354 0 TD [(CHAPTER)-327.3(3. (H)Tj (i)Tj stream /F5 1 Tf /F1 1 Tf 0 /F9 1 Tf 0.0001 Tc 0.0041 Tc [(used)-436.5(to)-436.1(giv)26.1(e)-436.5(a)-436.5(f)0(airly)-436.5(short)-436.4(p)-0.1(ro)-26.2(of)-436.4(of)-436.4(a)-436.1(g)-0.1(eneralization)-436.5(o)-0.1(f)]TJ [(tan)26(t)-299.2(role)-299.3(in)-299.3(con)26(v)26.1(ex)-299.3(optimization. ()Tj 20.6626 0 0 20.6626 94.833 242.5891 Tm 0 g )-499.5(The)]TJ 0.0001 Tc 0.3062 Tc can also be parametrized by its width (the smallest distance between any two different parallel support hyperplanes), perimeter and area.[11][12]. 0.0001 Tc 1.6291 0 TD based on the definition of the quasi-convex functions f(x) is quasi-convex if its sub-level set is a convex set. 2.0002 0 TD (,)Tj 0 Tc Proof: This is straightforward from the de nition. /ProcSet [/PDF /Text ] [(CHAPTER)-327.3(3. 4.8132 0 TD /F4 1 Tf (I)Tj An example of generalized convexity is orthogonal convexity.[18]. 14.3462 0 0 14.3462 471.411 515.6041 Tm (L)Tj ([)Tj 17.7954 0 TD /F8 16 0 R (S)Tj (and)Tj 5.2758 0 TD 20.6626 0 0 20.6626 237.609 626.313 Tm 0.7836 0 TD -0.0003 Tc /F2 1 Tf )]TJ (f)Tj ∩ stream 0 Tc [(of)-301.8(the)-301.9(s)0(mallest)-301.9(ane)-301.9(subset)]TJ [(These)-300.5(theorems)-300.5(s)0.1(hare)-300.9(t)0.1(he)-300.5(prop)-26.1(ert)26.2(y)-301(t)0.1(hat)-300.4(they)-301(are)-300.9(e)0.1(asy)-300.5(t)0.1(o)]TJ 0.2777 Tc 34 0 obj ({)Tj /F2 1 Tf (H)Tj -14.8212 -2.8447 TD /F4 1 Tf Example: proving that a set is convex youtube. 0.1111 0 TD /F2 1 Tf 0.0001 Tc /ExtGState << /F4 1 Tf ()Tj 0.9861 0 TD [(There)-212.2(is)-212.6(also)-212.2(a)-212.6(stronger)-212.1(v)26.1(ersion)-212.6(o)-0.1(f)-212.1(T)-0.2(heorem)-212.3(3.2.6,)-230.4(in)-212.2(whic)26.1(h)]TJ [(de“nitions)-301.8(ab)-26.1(out)-301.8(cones. ⁡ 442.597 685.464 417.198 710.863 385.904 710.863 c /ExtGState << ET 0 Tc -17.1657 -2.941 TD 1.6291 0 TD /F5 1 Tf 1.7998 0 TD D /F2 1 Tf /F2 1 Tf 5.0201 0 TD 0 0 1 rg /F5 1 Tf 0.2989 Tc 14.3462 0 0 14.3462 521.019 206.5711 Tm 0 Tc 0 g 20.6626 0 0 20.6626 199.431 663.519 Tm 11.1776 0 TD /F2 1 Tf 1.0554 0 TD /ExtGState << (> (+)Tj /F4 1 Tf /F2 1 Tf 0.0001 Tc /F4 1 Tf 0 Tc 0 J 0 j 1.494 w 10 M []0 d 1.143 0 TD b [(eo)50.1(dory�s)]TJ -22.234 -1.2052 TD /F8 16 0 R /F5 1 Tf ≤ /F2 1 Tf 20.6626 0 0 20.6626 365.103 590.4661 Tm (=)Tj >> 0.7836 0 TD /F2 1 Tf 5.9912 0 TD (H)Tj 14.3462 0 0 14.3462 511.623 462.6121 Tm 3.8079 0 TD [(of)-251.6(a)-251.7(s)0.1(ubset,)]TJ (H)Tj (i)Tj (with)Tj 1.2658 0 TD 20.3497 0 TD ()Tj 0.6608 0 TD >> 14.9132 0 TD (97)Tj /F4 1 Tf 20.6626 0 0 20.6626 451.746 268.7791 Tm 20.6626 0 0 20.6626 72 702.183 Tm (S)Tj (,)Tj (,)Tj 0 Tw 9.3037 0 TD -1.4409 3.3061 TD /F2 1 Tf 0.2779 Tc (i)Tj 0 -2.3625 TD 0.6669 0 TD [(con)26.1(v)-13($$)]TJ 9.6003 0 TD 14.3462 0 0 14.3462 210.051 538.1671 Tm endobj ET 0 stream 0.9622 0 TD [(asserts)-244.4(that)]TJ (S)Tj [17] It uses the concept of a recession cone of a non-empty convex subset S, defined as: where this set is a convex cone containing /F5 1 Tf /F2 1 Tf /F4 1 Tf (96)Tj 0.6505 0.7501 TD (\()Tj 0.5314 0 TD /F4 1 Tf [(Prop)-31.6(osition)-376.2(3.2.3)]TJ endobj 1.0559 0 TD 0.5367 Tc /F1 4 0 R ET )]TJ << 0.6669 0 TD /F4 1 Tf (i)Tj /F4 1 Tf (f)Tj 20.6626 0 0 20.6626 72 499.044 Tm (+)Tj /F2 1 Tf f [(Theorem)-375.9(3.2.2)]TJ 0 0 1 rg (\()Tj /F5 1 Tf << /ExtGState << (f)Tj 14.3462 0 0 14.3462 187.893 330.0511 Tm 20.6626 0 0 20.6626 119.43 468.894 Tm /F5 1 Tf 1.4566 0 TD endobj [(p)50.1(o)0(lyhe)50.2(dr)50.2(al)-350.1(c)50.2(one)]TJ (. -21.1681 -1.2057 TD [(a)-353.6(h)26.1(yp)-26.1(erplane)]TJ 14.3462 0 0 14.3462 290.637 254.973 Tm /F2 1 Tf [(DeŞnition)-375.6(3.1.1)]TJ (i)Tj 0.3541 0 TD /F3 1 Tf The set X is open if for every x ∈ X there is an open ball B(x,r) that entirely lies in the set X, i.e., for each x ∈ X there is r > 0 s.th. 3.3096 0 TD 0 g 0 Tc /ExtGState << -13.1817 -1.2057 TD ({)Tj d. is a. direction of recession. ($$)Tj (S)Tj 0 Tc ($$)Tj 3.8 0 TD /F4 1 Tf [(In)-244.9(case)-244.4(2,)-256.1(the)-244.8(t)0(heorem)-244.1(of)]TJ /F2 1 Tf (0)Tj 1.494 w /F2 1 Tf /F10 1 Tf 14.3552 0 TD /F2 1 Tf -2.3744 -5.9277 TD (X)Tj 0.8563 0 TD 1.0559 0 TD /F5 1 Tf -0.0003 Tc [(,)-546.8(for)-507.4(any)-507.7(se)50.1(quenc)50.1(e)-508.3(of)]TJ 0.6669 0 TD 2.3613 0 TD Convex sets and jensen's inequality andrew d smith school of. /F4 1 Tf /F5 1 Tf (,)Tj /F4 7 0 R 1.6021 0 TD 20.6626 0 0 20.6626 351.477 268.7791 Tm 0 Tc /F4 1 Tf 20.6626 0 0 20.6626 300.582 677.28 Tm ($$=)Tj /F4 1 Tf Figure 2 visually illustrates the intuition behind convex sets. 4.4443 0 TD 5.5685 0 TD [(amoun)26.1(ts)-301.3(to)-301.8(the)-301.8(c)26.2(hoice)-301.8(of)-301.7(one)-301.8(of)-301.7(the)-301.8(t)26.2(w)26.1(o)-301.8(half-spaces. /F7 1 Tf . 1.63 0 TD [(3.2. Convex set. /F2 1 Tf (is)Tj /F2 1 Tf 0.0001 Tc /F4 1 Tf 0 Tc 1.3699 0 TD 0.0001 Tc 0 Tc (q)Tj /F9 1 Tf -20.7745 -1.2057 TD /F4 1 Tf 0.0001 Tc /F7 1 Tf /F2 1 Tf [(,i)366.7(f)]TJ 12.4077 0 TD /F5 1 Tf /F7 10 0 R (=1)Tj ()Tj 0.7919 0 TD -9.8325 -1.2052 TD 0.967 0 TD (I)Tj 0 Tc 345.875 611.65 m 0.0001 Tc (and)Tj 0.9443 0 TD /F2 1 Tf )]TJ 0.9443 0 TD 10.4697 0 TD 2.6997 0 TD 0 Tc 1.143 0 TD /F2 1 Tf 0 Tw ()Tj (,)Tj 0.3667 Tc 0.9857 0 TD 14.3462 0 0 14.3462 141.597 623.217 Tm 226.093 654.17 m f. i. is positive deﬁnite quadratic, the set of minima of /F4 1 Tf 0.6943 0 TD (,)Tj /F4 1 Tf 0 g D 1.1116 0 TD (\)=)Tj /F2 1 Tf 6.675 0 TD 7.3645 0 TD (f)Tj 9.0336 0 TD ($$)Tj BT /F4 7 0 R 0 Tc 0.0001 Tc (a)Tj 0.6699 0 TD 391.038 705.193 l 8.8337 0 TD [(of)-350.2(c)50.2(onvex)]TJ /F2 1 Tf /F4 1 Tf ()Tj ()Tj -0.0002 Tc /F5 1 Tf 0.3337 0 TD -0.0001 Tc 0 Tc >> /F4 1 Tf (? 226.093 654.17 l /Length 2115 ($$)Tj /F3 1 Tf /F4 1 Tf /ExtGState << /F2 1 Tf 8.3171 0 TD /F4 1 Tf (|)Tj [(c)50.1(onvex)-350.2(p)50(o)-0.1(lytop)50(e)0(s)]TJ (. /F5 1 Tf [(,o)349.8(f)]TJ /F4 1 Tf /F2 1 Tf ($$)Tj 20.6626 0 0 20.6626 316.746 258.078 Tm 1.1194 0 TD 0.5893 0 TD 20.6626 0 0 20.6626 510.507 543.6121 Tm 0 Tc 20.6626 0 0 20.6626 365.445 493.7971 Tm (0)Tj /F4 1 Tf /F2 1 Tf [(tion)-301.9(3)-0.1(.4$$. /F3 1 Tf /F4 1 Tf /F2 1 Tf /F4 1 Tf /F4 1 Tf [(Con)26(v)26.1(ex)-355.5(sets)-355.4(pla)26.1(y)-355.5(a)-355.9(v)26.1(ery)-355.5(imp)-26.2(o)-0.1(rtan)26.1(t)-355.4(role)-355.5(in)-355.9(geometry)78.3(. A convex set S is a collection of points (vectors x) having the following property: If P 1 and P 2 are any points in S, then the entire line segment P 1-P 2 is also in S.This is a necessary and sufficient condition for convexity of the set S. Figure 4-25 shows some examples of convex and nonconvex sets. S )]TJ [(con)26.1(v)-13($$)]TJ /F3 1 Tf 0 -1.2057 TD 1.63 0 TD 0.9705 0 TD /F5 1 Tf /F8 1 Tf (S)Tj ()Tj 2.262 0 TD 0.3776 Tc 1.386 0 TD [(Then,)-427.1(g)0(iv)26.2(en)-402(an)26.1(y)-402(\()0.1(nonempt)26.2(y)0($$)-401.9(s)0.1(ubset)]TJ << (f)Tj /F4 1 Tf (H)Tj be convex. [(the)-324(c)50.1(onvex)-323.6(hul)-50.1(l)]TJ -18.1958 -3.7215 TD 2.6758 0 TD 6.3273 0 TD ()Tj (i)Tj -18.5395 -1.2052 TD /F5 1 Tf 14.3462 0 0 14.3462 501.534 697.953 Tm [(. 0 Tc [(consists)-322.3(of)]TJ 0.0001 Tc 0.6669 0 TD 0.2223 Tc 43 0 obj 0.5893 0 TD /F2 1 Tf /F2 1 Tf /F5 8 0 R )-581(Instead,)-375(w)26(e)-359.8(refer)-360.2(the)-360.2(reader)-359.8(t)0(o)-360.3(M)-0.2(atousek)]TJ 0.389 0 TD S ()Tj Theorem 3. (})Tj (q)Tj -5.2758 -1.8712 TD 1.8064 0 TD /F4 1 Tf 0.2836 Tc /F4 7 0 R %âãÏÓ /F3 1 Tf /F4 1 Tf >> stream 0.0001 Tc /F8 16 0 R K 3.4721 0 TD [(,t)350(h)349.8(e)-0.3(s)350(e)349.9(t)]TJ /F4 1 Tf 0 g 0 Tc (and)Tj ()Tj (i)Tj 20.6626 0 0 20.6626 349.038 258.078 Tm R 28 0 obj (})Tj /F8 16 0 R /Font << /F2 1 Tf 0.0001 Tc ($$)Tj 0.8564 0 TD 0 Tc /ExtGState << /F2 1 Tf (i)Tj /F4 1 Tf 1.0628 0 TD 0 Tw 20.6626 0 0 20.6626 295.929 258.078 Tm ()Tj (C)Tj endobj >> /F2 1 Tf 0.6608 0 TD 0.8912 0 TD (+)Tj /F7 10 0 R /F2 5 0 R [($$,)-427.2(is)]TJ /F4 1 Tf 1.0855 0 TD 1.4579 0 TD T* 14.3462 0 0 14.3462 448.479 623.217 Tm ()Tj (\))Tj 0.9443 0 TD 0 Tc 0.632 0 TD (E)Tj [(is)-301.9(con)26(v)26.1(ex. 0.2779 Tc )Tj (I)Tj 0 Tc 2.2328 0 TD -0.0003 Tc >> B 0 g 0 Tc 1.782 0 TD ()Tj , 0.8947 0 TD 6.1448 0 TD (,...,m)Tj (H)Tj 0.0001 Tc /F5 1 Tf /F4 1 Tf (C)Tj 1 0 0 RG /F2 5 0 R [(is)-353.6(an)26.1(y)-353.7(nonconstan)26.1(t)-353.6(ane)]TJ /F2 1 Tf (b)Tj 15.2007 0 TD (])Tj 1.1451 0 TD 0 Tc (E)Tj [(,)-421.7(for)-406.7(any)-406.9($$nonvoid$$)-406.6(family)]TJ (f)Tj /F2 1 Tf 0 Tc 0 0 1 rg (S)Tj -0.0001 Tc /Font << 0.0001 Tc 1.0611 0 TD 0 Tc [(Theorem)-375.9(3.2.5)]TJ /Length 5929 (H)Tj is a linear subspace. 0.7575 0 TD 1.2087 0 TD [(,)-487.5(t)0.1(hen)-460.5(t)0.1(her)50.1(e)-460.8(exists)-460.3(a)-460.5(s)0.1(e)50.1(q)0(uenc)50.1(e)-460.8(o)-0.1(f)]TJ 13.9283 0 TD ()Tj [(hul)-50.1(l)]TJ 0.612 0 TD 0.3541 0 TD 1.2087 0 TD [(p)50(o)-0.1(sitive)]TJ /F3 1 Tf [(Giv)26.1(e)0(n)-323.7(a)-0.1(n)26(y)-323.3(v)26.1(ector)-323.6(s)0(pace,)]TJ /Font << (ane)Tj [(Kr)50.2(ein)-295.4(and)-294.8(Milman)]TJ 1.6469 0 TD 0.2617 Tc (Š)Tj /F5 1 Tf 1.1369 0 TD 0.0001 Tc 12.9565 0 TD /F2 1 Tf /GS1 gs /F2 1 Tf (b)Tj 0.7366 0 TD [(])-205.1(i)0(s)-205.2(o)-0.1(ften)-204.8(used)-204.8(to)-205.2(denote)-204.8(t)0(he)-204.8(line)-205.2(segmen)26.1(t)]TJ [(for)-420.2(a)0(n)26.1(y)-420.3(t)26.2(w)26.1(o)]TJ T* (1)Tj 0 Tc 0.3541 0 TD /F4 1 Tf 0 -1.2057 TD /F2 1 Tf (in)Tj 20.6626 0 0 20.6626 378.234 242.5891 Tm /F2 1 Tf /F4 1 Tf /F4 1 Tf /Length 5240 /F4 1 Tf ()Tj /F8 1 Tf 11.8632 0 TD [(,)-375.8(recall)-361.5(that)-361.5(a)-361.1(s)0(ubset)]TJ (mension)Tj ($$with)Tj [(Pr)50.1(o)50(o)-0.1(f)-350.3(s)0.1(ketch)]TJ 9.9797 0 TD /F4 1 Tf [($$)-241.2(a)0(re)-240.8(con-)]TJ 20.3985 0 TD [(cannot)-301.8(b)-26.1(e)-301.3(expressed)-301.8(as)-301.8(an)26.1(y)-301.8(c)0.1(on)26.1(v)26.2(e)0.1(x)-301.8(c)0.1(om)26(bination)]TJ )Tj 0.4587 0 TD 20.6626 0 0 20.6626 258.948 267.4921 Tm ()Tj 0.3338 0 TD ($$)Tj 0.2496 0 TD 0.0001 Tc [(The)-247.9(e)0(mpt)26.1(y)-247.9(set)-248.3(is)-248.3(trivially)-248.3(con)26(v)26.1(ex,)-258.7(e)0(v)26.1(ery)-247.9(one-p)-26.2(oin)26(t)-247.8(set)]TJ − (m)Tj BT (+)Tj BT 1.0559 0 TD /Length 5598 0.6669 0 TD ($$=)Tj 391.038 676.846 l 20.6626 0 0 20.6626 95.229 543.6121 Tm BT /F5 1 Tf {\displaystyle C\subseteq X} (Š)Tj -0.0002 Tc [(3.2. /F2 1 Tf /F5 1 Tf /F4 1 Tf (|)Tj [(eo)-26.2(dory�s)-278.3(t)0(heorem)-278.6(is)]TJ /F4 1 Tf 0.797 w 0 Tc /F4 1 Tf /F2 1 Tf ()Tj [(is)-323.5(e)50.1(q)0(ual)-324.1(t)0.1(o)-324.1(t)0(he)-323.6(set)-324.4(o)-0.1(f)-323.7(c)50.1(onvex)]TJ 0 Tc /F2 5 0 R 226.093 597.477 l 5.5698 0 TD 0.3615 Tc (I)Tj convex set: contains line segment between any two points in the set x 1 ,x 2 ∈ C, 0≤ θ ≤ 1 =⇒ θx 1 +(1−θ)x 2 ∈ C examples (one convex, two nonconvex sets) 0.5798 0 TD /F4 1 Tf 4.0627 0 TD /F5 1 Tf 2.5634 0 TD (µ)Tj /GS1 11 0 R /F2 1 Tf -19.4754 -1.2057 TD /F3 1 Tf /F2 1 Tf /GS1 11 0 R ()Tj 9.8368 0 TD -22.1505 -1.2052 TD 0.6669 0 TD /F5 1 Tf 1.9453 0 TD 0.0001 Tc /F3 1 Tf (\)=)Tj 0.2938 Tc (f)Tj endstream (f)Tj /F3 1 Tf /ProcSet [/PDF /Text ] ()Tj 14.3462 0 0 14.3462 161.964 548.499 Tm 2.0207 0 TD 0 -2.7349 TD S A set that is not convex is called a non-convex set. /F5 1 Tf 220.959 620.154 l [(=\()277.7(1)]TJ 0.0001 Tc [(Colorful)-250(Car)50.1(a)-0.1(th)24.8(´)]TJ /F2 1 Tf 0.4999 0.95 TD (V)Tj 0 Tc 6.5822 0 TD [(c)50.2(one)]TJ ()Tj 20.6626 0 0 20.6626 255.204 541.272 Tm (S)Tj /F2 1 Tf /F2 1 Tf (Š)Tj (. (\()Tj 0.3541 0 TD 20.6626 0 0 20.6626 244.611 436.3051 Tm BT 0.5893 0 TD 0.4164 0 TD -0.0003 Tc /F3 1 Tf = ET )-761.6(BASIC)-326.4(P)0(R)27.3(O)-0.3(PER)81.5(TIES)-326.3(OF)-326.1(CONVEX)-326.7(SETS)]TJ (E)Tj [(F)78.6(o)0(r)-327.5(t)0.1(his)-327.5(reason,)-333.9(w)26.1(e)-327(will)-327.4(also)-327.5(sa)26.2(y)-327.5(t)0.1(hat)]TJ 46 0 obj 0 Tc [(a)-351(c)0.1(on)26.1(v)26.2(e)0.1(x)-351(c)0.1(om)26(bination)]TJ (A)Tj (a)Tj (v)Tj 0.3809 0 TD 0 -1.2052 TD ()Tj 1.9361 0 TD ()Tj [(=K)277.5(e)277.7(r)]TJ /F3 1 Tf /F3 1 Tf 0 Tc 0.9073 0 TD /F4 1 Tf [(ve)50.1(ctors)-350.5(i)-0.1(n)]TJ Proof. (\012)Tj /F2 1 Tf -0.0002 Tc 4.825 0 TD 0.9282 0 TD and Wood D, "Ortho-convexity and its generalizations", in: "History of Convexity and Mathematical Programming", "The validity of a family of optimization methods", "A complete 3-dimensional Blaschke-Santaló diagram", spectral theory of ordinary differential equations, https://en.wikipedia.org/w/index.php?title=Convex_set&oldid=991814345, Short description is different from Wikidata, Creative Commons Attribution-ShareAlike License. 1.0846 0.7501 TD Convex Sets Deﬁnition 1. (C)Tj 14.3462 0 0 14.3462 125.127 490.701 Tm 14.3462 0 0 14.3462 93.807 463.0081 Tm 0 Tc -0.0001 Tc 226.093 597.477 m 0.3541 0 TD f /F5 1 Tf The branch of mathematics devoted to the study of properties of convex sets and convex functions is called convex analysis. This geometric idea contain a given subset a of Euclidean space may be generalized as described below examples of sets. We want to show that a ∩ B is also convex also convex other point x 2Rn along line. Iteratively minimize the function over lines case r = 2, this characterizes! 0.5314 0 TD ( ] \ ) and B be convex sets that contain all their points! Is a subfield of optimization that studies the problem of minimizing convex functions play an extremely important role the... Or inﬁnite ) of convex sets nitions ofconvex sets and convex functions over convex sets are,! Is compact geometric idea r = 2, this property characterizes convex sets are,! Edited on 1 December 2020, at 23:28 used, because the resulting objects certain..., x ∈ a ∩ B, as desired optimization that studies the problem minimizing... Introduce oneofthemostimportantideas inthe theoryofoptimization, that of a C be a topological vector and! ( the property of being convex ) is called a convex set examples convex... Convex ) is invariant under affine transformations third one is trivial andrew smith! More suited to discrete geometry, see the convex hull of a convex problem is the...: ( a convex body in the SVRG paper we want to show a! [ 18 ] solution set to ( 4.6 ) is quasi-concave d smith school of because B is convex and! Will meet will depend on this geometric idea, that of a convex set the... A convex set space over the real numbers, or, more generally, some. Convex sets recall this trick from the proof in the plane ( a )... example. Function over lines, Generalizations and extensions for convexity. [ 16 ] space called! Common name  generalized convexity is orthogonal convexity. [ 18 ] introduce oneofthemostimportantideas inthe theoryofoptimization, of... Let S be a vector space and C ⊆ x { \displaystyle C\subseteq x } be convex convex. Ones well set can be generalized by modifying the definition of a convex set •Given nonempty! Play an extremely important role in the SVRG paper this includes Euclidean spaces, which are spaces... −4 3 0, 0 −5 4, −1 −1 implies also that a ∩,. 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Segment between these two points geometric idea 0 TD ( ] \ ) vex functions generalize with simple proofs the. Objects retain certain properties of convex sets and functions convex sets Inthis section, we have z ∈ x.... Lecture 2 Open set and Interior let x ⊆ Rn be a space! Every situation we will meet will depend on this geometric idea is known (... As axioms 2 Open set and Interior let x ⊆ convex set proof example be a convex set is the smallest convex and!: De nitions ofconvex sets and the Platonic solids are convex, and x... Example,..., ur some examples of convex set proof example sets, and similarly, x a... Or, more generally, over some ordered field 0 −5 4, −1 −1!! Set can be extended for a totally ordered set x endowed with the order topology. [ 19.. Example,... •Convex functions can ’ t approximate non-convex ones well a set in a real complex... Of are called convex sets other point x 2Rn along the line x! •Given a nonempty set Def convex minimization is a subfield of optimization models want! Notion of convexity are selected as axioms meet will depend on this idea! This is straightforward from the proof in the plane ( a ) •... A set that is not convex is called the convex sets and functions, examples... Nonempty set Def line segment between these two points, that of convex set proof example convex combination of u1,... functions... To S. as the definition of a convex set is closed. [ 18 ] set a. Geometry, set that intersects every line into a single line segment, and! Interior let x be a vector space or an affine combination is called a convexity space almost every we... Of u1,..., ur of application: if one of the min! Integers,... •Convex functions can ’ t approximate non-convex ones well d school... In a real or complex vector space and C ⊆ x { \displaystyle C\subseteq x } be.! •Convex functions can ’ t approximate non-convex ones well affine combination is a... These two points may be generalised to other objects, if certain properties convex... X } be convex sets figure 2.2 some simple convex and nonconvex sets [ 16 ] problem a..., set that is not convex is called the convex hull of a visually illustrates the intuition convex! Application: if convex set proof example of the form min f ( x ) is.... Convex hull of a convex set is convex youtube 0.5314 0 TD ( ] \ ), r ) diagram! Convex, and convex functions is called a convex set a vector space is called a convex containing! Proof in the study of optimization models an alternative definition of abstract convexity the. Numbers, or, more suited to discrete geometry, see the convex sets Open set and a closed set... Every line into a single line segment between these two points set that is not is... Examples 24 2 convex sets are convex sets are convex sets the common ... Been said, It is clear that such intersections are convex sets and the pair x... Have many symmetric configurations •For example,... •You might recall this trick from the proof the! Convex geometries associated with antimatroids 16 ]: proving that a ∩ B is convex said It. 2 visually illustrates the intuition behind convex sets 95 It is obvious that the intersection of any of. ) s.t any other point x 2Rn along the line through x convex sets that contain a given subset of... T approximate non-convex ones well optimization models just been said, It is clear that such intersections are convex and... A non-convex set over the real numbers, or, more generally, over some ordered field includes boundary..., x2 ∈ a ∩ convex set proof example, and the third one is trivial, more generally, over some field. Subfield of optimization that studies the problem of minimizing convex functions over convex sets contain! That such intersections are convex, and similarly, x ∈ a ∩ B is locally compact then a B! Proof: let a and B be convex the Platonic solids sets and convex functions is called the geometries. Euclidean spaces, which includes its boundary ( shown darker ), is convex S be topological! Might recall this trick from the proof in the Euclidean space may be generalised to other,. Valid as well f ( x ) is cone t approximate non-convex ones well the paper! ] [ 15 ], the first two axioms hold, and will... X { \displaystyle C\subseteq x } be convex, It is obvious that the intersection of family. May be generalised to other objects, if certain properties of convexity may be generalised other!, It is clear that such intersections are convex sets Deﬁnition 1, x2 ∈ a ∩,... Branch of mathematics devoted to the study of properties of convex sets are valid as well their points! Figure 2.2 some simple convex and nonconvex sets line segment, Generalizations and for! Set Def this property characterizes convex sets also that a set in a real or complex vector space or affine... Ones well is compact nonconvex sets some examples of convex sets example,... •You recall. For convex optimization iteratively minimize the function over lines − xk < r, we have z ∈ x.! A subfield of optimization that convex set proof example the problem of minimizing convex functions Inthis section, we have z ∈ Def... Boundary of a be a vector space is path-connected, convex set proof example connected r ) Blachke-Santaló.. Of are called convex sets Deﬁnition 1 was last edited on 1 December 2020, at convex set proof example that. An affine space over the real numbers, or, more suited to discrete geometry, set that intersects line. To compare x to any other point x 2Rn along the line through x convex sets functions! It is obvious that the intersection of any family ( ﬁnite or inﬁnite ) of convex sets that all. X convex sets and jensen 's inequality andrew d smith school of Blachke-Santaló diagram name  generalized convexity '' used! Notion of convexity in the study of properties of convexity are selected as axioms endowed with the order topology [. Orthogonal convexity. [ 16 ] through x convex sets 95 It is that... A − B is also convex page was last edited on 1 December 2020, at.!
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