Strongly convex space

Strongly convex spaces are normalized spaces considered in the mathematical subfield of functional analysis that meet a special convexity condition. This is a geometric property that has the consequence, among other things, that the edge of the unit sphere does not contain any "large", convex sets . This term goes back to Witold Lwowitsch Schmulian .

Strongly convex space: The non-empty intersection of sphere and convex set becomes arbitrarily small.

No strongly convex space: the non-empty intersection of the sphere and the convex set always has a positive diameter.

Definitions

For a normalized space, let the unit sphere and the sphere stretched by the factor , that is, the sphere around 0 with a radius . For a subset, let the diameter of this set and the distance between a point and this set. ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle B_{X}:=\{x\in X;\,\|x\|\leq 1\}}$ ${\displaystyle rB_{X}}$ ${\displaystyle r>0}$ ${\displaystyle r}$ ${\displaystyle A\subset X}$ ${\displaystyle \mathrm {diam} (A):=\sup\{\|x-y\|;\,x,y\in A\}}$ ${\displaystyle \mathrm {dist} (A,y):=\inf\{\|x-y\|;\,x\in A\}}$ ${\displaystyle y}$

A normalized space is called strongly convex if for every non-empty convex set : ${\displaystyle (X,\|\cdot \|)}$ ${\displaystyle C\subset X}$

${\displaystyle \mathrm {diam} (rB_{X}\cap C)\rightarrow 0}$for . ${\displaystyle r\searrow \mathrm {dist} (C,0)}$

Examples and characteristics

• As the adjacent drawings make clear, the one with the Euclidean norm is strongly convex, but not with the sum norm . This also shows that strong convexity depends on the norm and not just on the isomorphism class of the space. ${\displaystyle \mathbb {R} ^{2}}$
• Uniformly convex spaces are strongly convex, strongly convex spaces are strictly convex , the inversions generally do not hold.
• According to Ky Fan and Irving Glicksberg , every strongly convex space has the Radon-Riesz property and, conversely, every reflexive , strictly convex space with the Radon-Riesz property is strongly convex.
• Let it be the sequence space of the absolutely summable sequences with the norm as well as the sequence space of the quadratically summable sequences with the norm . As is known, and by a to be equivalent to standard defined. Then it is strictly convex, has the Radon-Riesz property (even the stronger Schur property ), but is not strongly convex. ${\displaystyle \ell ^{1}}$ ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle \ell ^{2}}$ ${\displaystyle \|\cdot \|_{2}}$ ${\displaystyle \ell ^{1}\subset \ell ^{2}}$ ${\displaystyle \textstyle \|x\|:=(\|x\|_{1}^{2}+\|x\|_{2}^{2})^{1/2}}$ ${\displaystyle \|\cdot \|_{1}}$ ${\displaystyle \ell ^{1}}$ ${\displaystyle (\ell ^{1},\|\cdot \|)}$

Equivalent characterizations

It turns out that one does not have to consider all convex sets of normalized space in the definition of strong convexity ; it is sufficient to restrict oneself to closed half-spaces . As is well known, this can be described by the real parts of continuous, linear functionals , i.e. by elements of the dual space . This is reflected in the following list of equivalent statements about a normalized space : ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X}$

• ${\displaystyle X}$ is strongly convex.
• For each applies to . ${\displaystyle f\in X',\|f\|=1}$ ${\displaystyle \mathrm {diam} \{x\in X;\,\|x\|\leq 1,\mathrm {Re} f(x)\geq 1-\delta \}\rightarrow 0}$ ${\displaystyle \delta \searrow 0}$
• If a sequence is in with for all members of the sequence and is with , then the sequence is a Cauchy sequence . ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle f\in X',\|f\|=1}$ ${\displaystyle \mathrm {Re} f(x_{n})\rightarrow 1}$
• If it is not empty and convex, and a sequence in with , then the sequence is a Cauchy sequence. ${\displaystyle C\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle (y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle C}$ ${\displaystyle \|x-y_{n}\|\rightarrow \mathrm {dist} (C,x)}$

The Cauchy sequences in the above equivalent characterizations are generally not convergent due to a lack of completeness . Taking the completeness into account, the following statements are equivalent for a normalized space : ${\displaystyle X}$

• ${\displaystyle X}$is a strongly convex Banach space .
• If a sequence is in with for all members of the sequence and is with , then the sequence converges. ${\displaystyle (x_{n})_{n\in \mathbb {N} }}$ ${\displaystyle X}$ ${\displaystyle \|x_{n}\|=1}$ ${\displaystyle f\in X',\|f\|=1}$ ${\displaystyle \mathrm {Re} f(x_{n})\rightarrow 1}$
• If not empty, closed and convex, and a sequence in with , then the sequence in converges . ${\displaystyle C\subset X}$ ${\displaystyle x\in X}$ ${\displaystyle (y_{n})_{n\in \mathbb {N} }}$ ${\displaystyle C}$ ${\displaystyle \|x-y_{n}\|\rightarrow \mathrm {dist} (C,x)}$ ${\displaystyle C}$
• ${\displaystyle X}$is reflexive, strictly convex and has the Radon-Riesz property.

Individual evidence

1. ^ VL Schmulian: Sur la dérivabilité de la norme dans l'espace de Banach , Doklady Acad. Sci. URSS (1940) vol. 27, pp. 643-648
2. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , definition 5.3.15
3. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , sentence 5.3.16
4. ↑ K. Fan, I. Glicksberg: Some geometric properties of the speres in a normed space , Duke Math. J (1958), Volume 25, pages 553-568
5. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , theorems 5.3.22 and 5.3.23
6. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , theorems 5.3.17 and 5.3.20
7. ^ Robert E. Megginson: An Introduction to Banach Space Theory. Springer-Verlag, 1998, ISBN 0-387-98431-3 , Corollary 5.3.18, Theorem 5.3.21, and 5.33