Theorema bipolare

Theorema bipolare in mathematica est theorema in explicatione convexa quod condiciones necessarias et satis adhibet ut conus aequet suum bipolare. Theorema bipolare videri potest proprius theorematis Fenchel-Moreauani casus.

Pronuntiatum theorematis

In quaque copia non vacua, $C\subset X$ in nonnullo spatio lineari $X$, tum conus bipolaris $C^{oo}=(C^o{)}^o$ datur a

$C^{oo}=\mathrm{cl}(\mathrm{co}\{\lambda c:\lambda \ge 0,c\in C\})$

ubi $\mathrm{co}$ corticem convexum denotat.

Casus proprius

$C\subset X$ est non vacuus conus convexus clausus si et solum si $C^{++}=C^{oo}=C$ cum $C^{++}=(C^{+}{)}^{+}$, ubi $(\cdot {)}^{+}$ denotat conum dualem positivum.

Generatim, si $C$ sit conus convexus, tum conus bipolaris datur a

$C^{oo}=\mathrm{cl}C.$

Coniunctio cum theoremate Fenchel–Moreauano

Si $f(x)=\delta (x|C)=\{\begin{array}{cc}0& \text{if }x\in C\\ +\mathrm{\infty }& \text{else}\end{array}$ sit functio propria coni $C$, tum coniugatum convexum $f^{*}(x^{*})=\delta (x^{*}|C^o)={\delta }^{*}(x^{*}|C)=\underset{x\in C}{\sup }⟨x^{*},x⟩$ est functio firmamenti pro $C$, et $f^{**}(x)=\delta (x|C^{oo})$. Ergo, $C=C^{oo}$ si et solum si $f=f^{**}$.