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Consider the following properties of a Banach space:

the intersection of any support hyperplane with the unit sphere is

(S) a singleton (this is the strict convexity);

(SF) finite-dimensional set;

(SC) compact in the norm topology.

It is easy to see that these properties are equivalent to the fact that any closed convex subsets of a unit sphere is s ingleton/finite-dimensional/ compact.

Q1: Are (SF) and (SC) different?

Q2: Were these conditions considered in the literature? So they have names?

Q3: Are there any necessary or sufficient conditions for them? In particular, are there any dual/predual conditions?

Consider the following properties of a Banach space:

the intersection of any support hyperplane with the unit sphere is

(S) a singleton (this is the strict convexity);

(SF) finite-dimensional set;

(SC) compact in the norm topology.

It is easy to see that these properties are equivalent to the fact that any closed convex subset of a unit sphere is singleton/finite-dimensional/ compact.

Q1: Are (SF) and (SC) different?

Q2: Were these conditions considered in the literature? Do they have names?

Q3: Are there any necessary or sufficient conditions for them? In particular, are there any dual/predual conditions?

Consider the following property of a Banach space $E$: intersection of any support hyperplane with the unit sphere is compact in the norm topology (recall that if we replace "compact" with "singleton" we get strict convexity).

IT is easy to see that this is equivalent to the fact that any convex subsets of a unit sphere is compact.

Does this condition have a name? Is there any necessary or sufficient condition for it? In particular, is there any dual/predual condition?

Is it actually possible that these intersections are not finite-dimensional convex sets?

Consider the following properties of a Banach space:

the intersection of any support hyperplane with the unit sphere is

(S) a singleton (this is the strict convexity);

(SF) finite-dimensional set;

(SC) compact in the norm topology.

It is easy to see that these properties are equivalent to the fact that any closed convex subsets of a unit sphere is s ingleton/finite-dimensional/ compact.

Q1: Are (SF) and (SC) different?

Q2: Were these conditions considered in the literature? So they have names?

Q3: Are there any necessary or sufficient conditions for them? In particular, are there any dual/predual conditions?

Consider the following property of a Banach space $E$: intersection of any hyperplane with the unit sphere is compact in the norm topology (recall that if we replace "compact" with "singleton" we get strict convexity).

Does this condition have a name? Is there any necessary or sufficient condition for it? In particular, is there any dual/predual condition?

Is it actually possible that these intersections are not finite-dimensional convex sets?

Consider the following property of a Banach space $E$: intersection of any support hyperplane with the unit sphere is compact in the norm topology (recall that if we replace "compact" with "singleton" we get strict convexity).

IT is easy to see that this is equivalent to the fact that any convex subsets of a unit sphere is compact.

Does this condition have a name? Is there any necessary or sufficient condition for it? In particular, is there any dual/predual condition?

Is it actually possible that these intersections are not finite-dimensional convex sets?