Extreme points and polar sets

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Problem 50.23 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 50 "Convex sets"

Let MM be a compact convex set containing a neighborhood of zero, in a vector space VV regarded as an affine space, and let

M∗={f∈V∗∣f(x)⩽1 for every x∈M}. M^{*}=\left\{ f \in V^{*} \mid f(x) \leqslant 1 \text{ for every } x \in M\right\} .

Prove that:

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(a)

M∗M^{*} is a compact convex set in V∗V^{*} containing a neighborhood of zero;

(b)

M∗∗=MM^{* *}=M under the canonical identification of V∗∗V^{* *} with VV.

Problem 50.24 from Кострикин А.И. "Сборник задач по алгебре" [Кос15]More problems in Section 50 "Convex sets"

Prove that every compact convex set coincides with the convex hull of the set of its extreme points.

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References

[Кос15] Кострикин А.И. Сборник задач по алгебре. МЦНМО, 2015.