A strictly convex space is a normed vector space $(X,|\cdot|)$ whose unit ball is a strictly convex set. Equivalently, the norm satisfies the following strict convexity condition:
$$ |x| = |y| = 1,; x eq y ;\Longrightarrow; \Big|\frac{x+y}{2}\Big| < 1 . $$
In other words, any non‑trivial convex combination of distinct points on the unit sphere lies in the interior of the unit ball. This property is sometimes described by saying that the unit sphere contains no non‑degenerate line segments.
Equivalent characterizations
- Rotundity: A strictly convex space is also called a rotund Banach space. Rotundity is precisely the geometric formulation above.
- Uniqueness of best approximations: In a strictly convex space, for any closed convex subset $C \subset X$ and any point $x \in X$, there is at most one point of $C$ that attains the minimal distance $\inf_{c\in C}|x-c|$. Hence best approximations, when they exist, are unique.
- Extreme points of the unit ball: Every point of the unit sphere is an extreme point of the unit ball.
Relation to other convexity notions
- Uniform convexity: Uniform convexity is a stronger condition; every uniformly convex space is strictly convex, but the converse need not hold. Uniform convexity imposes a quantitative modulus of convexity that controls how far the midpoint of two unit vectors can be from the unit sphere as a function of the distance between the vectors.
- Smoothness: Strict convexity concerns the shape of the unit ball, whereas smoothness (or Gateaux differentiability of the norm) concerns the uniqueness of supporting hyperplanes at points of the unit sphere. The two properties are independent: a space may be strictly convex but not smooth, smooth but not strictly convex, both, or neither.
Examples
- Hilbert spaces: Every Hilbert space $(H,|\cdot|_2)$ is strictly convex because the parallelogram law yields equality in the strict convexity condition only when the vectors are identical.
- $L^p$ spaces: For $1 < p < \infty$, the Lebesgue spaces $L^p(\mu)$ are strictly convex. For $p=1$ or $p=\infty$ strict convexity fails, as the unit ball contains flat faces.
- $\ell^p$ sequence spaces: The sequence spaces $\ell^p$ are strictly convex for $1 < p < \infty$ and not strictly convex for $p=1$ or $p=\infty$.
- Finite‑dimensional normed spaces: Any norm on $\mathbb{R}^n$ whose unit ball is a smooth, round shape (e.g., Euclidean norm) yields a strictly convex space; polyhedral norms (e.g., the $\ell^1$ norm) do not.
Applications
Strict convexity is pivotal in approximation theory, optimization, and functional analysis:
- Uniqueness of minimizers: Strict convexity of the norm ensures uniqueness of solutions to problems of minimal norm, such as projection onto convex sets and best‑approximation problems.
- Fixed‑point theorems: Certain fixed‑point results (e.g., Browder–Kirk theorem) assume strict convexity of the underlying Banach space.
- Geometric functional analysis: Strict convexity interacts with duality; the dual of a strictly convex space is smooth, and the dual of a smooth space is strictly convex.
Historical notes
The concept of strict convexity of normed spaces originated in the early development of functional analysis in the 1930s and 1940s, paralleling the study of convex sets in geometry. The term “rotund” was introduced by J. A. Clarkson in 1936 in his seminal paper on uniformly convex spaces, where he also proved that reflexive Banach spaces admit equivalent strictly convex norms. Subsequent work by James, Day, and others refined the relationships among strict convexity, smoothness, and duality.
Remarks
- Strict convexity is a purely geometric property of the norm; two equivalent norms on the same vector space may differ in whether they are strictly convex.
- The property is preserved under taking closed subspaces and under forming $\ell^p$-sums with $1 < p < \infty$. It is not, in general, preserved under arbitrary linear topological isomorphisms unless the isomorphism is an isometry.