Opukloyu mnozhinoyu v evklidovomu abo afinnomu prostori nazivayetsya taka mnozhina yaka razom z dovilnimi dvoma tochkami sho nalezhat mnozhini maye u sobi vidrizok sho yih z yednuye 1 Opukla mnozhina viglyadaye yak deformovane kolo Chornij vidrizok z yednuye tochki x ta y i roztashovanij povnistyu v zelenij mnozhini Oskilki ce vikonuyetsya dlya bud yakih tochok x ta y mnozhini to mnozhina bude opukloyu Priklad neopukloyi mnozhini Tak yak chervona chastina chorne ta chervone vidrizku sho z yednuye tochki x ta y roztashovana za mezhami zelenoyi mnozhini to mnozhina ne bude opukloyu Zmist 1 Viznachennya 2 Prikladi 3 Vlastivosti opuklih mnozhin 4 Div takozh 5 Posilannya 6 LiteraturaViznachennya RedaguvatiInshimi slovami mnozhina X R n displaystyle X subset mathbb R n nbsp nazivayetsya opukloyu yaksho dlya tochok x 1 x 2 X displaystyle x 1 x 2 in X nbsp sho zadayutsya radius vektorami r 1 r 2 displaystyle vec r 1 vec r 2 nbsp tochka a r 1 1 a r 2 X a 0 1 displaystyle alpha vec r 1 1 alpha vec r 2 in X alpha in 0 1 nbsp Tobto mnozhina X displaystyle X nbsp razom z bud yakimi dvoma tochkami x 1 x 2 displaystyle x 1 x 2 nbsp yaki nalezhat cij mnozhini mistit vidrizok yakij yih z yednuye x 1 x 2 x x r 1 a r 2 r 1 a 0 1 displaystyle x 1 x 2 left x colon x vec r 1 alpha vec r 2 vec r 1 alpha in 0 1 right nbsp Prikladi RedaguvatiU prostori R 1 displaystyle mathbb R 1 nbsp opuklimi mnozhinami budut tochka vidrizok interval promin pryama U prostori R n displaystyle mathbb R n nbsp opuklim bude sam prostir bud yakij jogo linijnij pidprostir kulya opukli mnozhini prostoriv menshoyi vimirnosti Takozh opuklimi budut taki mnozhini pryama l x 0 h displaystyle l x 0 h nbsp sho prohodit cherez tochku x 0 displaystyle x 0 nbsp v napryamku vektora h displaystyle h nbsp l x 0 h x R n x x 0 a h a R n displaystyle l mathbf x 0 h left x in mathbb R n x x 0 alpha h alpha in mathbb R n right nbsp promin l x 0 h displaystyle l x 0 h nbsp yakij vihodit iz tochki x 0 displaystyle x 0 nbsp v napryamku vektora h displaystyle h nbsp l X 0 h x R n x x 0 a h a 0 displaystyle l mathbf X 0h left x in mathbb R n x x 0 alpha h alpha geqslant 0 right nbsp giperploshina Hpb z normallyu p H p b x R n p x b displaystyle mathrm H p beta left x in mathbb R n p x beta right nbsp pivprostori na yaki giperploshina podilyaye prostir H p b x R n p x b displaystyle mathrm H p beta left x in mathbb R n p x geqslant beta right nbsp H p b x R n p x b displaystyle mathrm H p beta left x in mathbb R n p x leqslant beta right nbsp Vsi perelicheni mnozhini krim kuli ye chastkovimi vipadkami opukloyi mnozhini poliedru Chotirikutnik na ploshini mozhe buti opuklim i neopuklim Vlastivosti opuklih mnozhin RedaguvatiPeretin opuklih mnozhin ye opuklim Linijna kombinaciya tochok opukloyi mnozhini opukla Opukla mnozhina mistit bud yaku opuklu kombinaciyu svoyih tochok Bud yaku tochku n vimirnogo evklidovogo prostoru z opukloyi obolonki mnozhini mozhna predstaviti yak opuklu kombinaciyu ne bilsh nizh n 1 tochok ciyeyi mnozhini Div takozh Redaguvati nbsp Portal Matematika Zadacha opuklogo programuvannya Opukla obolonka Opuklij politop Opukla poverhnya Lema Shepli Folkmana Opuklij analizPosilannya Redaguvati Analitichna geometriya Navch posibnik dlya stud mat spec un tiv per s rus O A Borisenko L M Ushakova Per G Ch Kurinnij Harkiv Osnova 1993 192 s Literatura RedaguvatiPolovinkin E S Balashov M V Elementy vypuklogo i silno vypuklogo analiza M FIZMATLIT 416 s ISBN 5 9221 0499 3 nbsp Ce nezavershena stattya z matematiki Vi mozhete dopomogti proyektu vipravivshi abo dopisavshi yiyi Otrimano z https uk wikipedia org w index php title Opukla mnozhina amp oldid 35488737
