The first characterization is related to Banach space theory and metric geometry. We say that a normed space (R^n,Vert cdot Vert ) is uniformly convex (or,more precisely, 2-uniformly conver) if there is a constant Cgeqslant 1 such that Vert (v+w)/(2)Vert ^2leqslant (1)/(2)Vert vVert ^2+(1)/(2)Vert wVert ^2-(1)/(4C)Vert w-vVert ^2 (1.2) for all v, win R^n . Then we have, by recursive applications of (1.2) and the continuity of the norm, the inequality Vert (1-t)v+twVert ^2leqslant (1-t)Vert vVert ^2+tVert wVert ^2-((1-t)t)/(C)Vert w-vVert ^2 (1.3) for all v,win R^n and tin [0,1] Observe that what simply follows from the convexity of the norm is Vert (1-t)v+tw)Vert ^2leqslant (1-t)Vert vVert ^2+tVert wVert ^2 and (1.3) means that Vert cdot Vert possesses a stronger convexity.

The provided text describes a property of normed spaces called uniform convexity. Let's break down the key concepts and the implications of inequalities (1.2) and (1.3).

1. Normed Spaces: A normed space is a vector space (like Rⁿ) equipped with a norm, denoted ||·||. The norm assigns a non-negative length to each vector, satisfying certain properties (e.g., ||v|| = 0 if and only if v = 0, ||cv|| = |c| ||v|| for any scalar c, and the triangle inequality ||v + w|| ≤ ||v|| + ||w||).

2. Uniform Convexity: A normed space is uniformly convex if there exists a constant C ≥ 1 such that inequality (1.2) holds for all vectors v and w in the space. This inequality strengthens the usual convexity property of the norm. Let's analyze it:

* The left-hand side: ||(v+w)/2||² represents the squared norm of the midpoint between vectors v and w.
* The right-hand side: (1/2)||v||² + (1/2)||w||² is the average of the squared norms of v and w.
* The crucial term: -(1/4C)||w - v||² is a penalty term proportional to the squared distance between v and w. The presence of this term is what distinguishes uniform convexity from ordinary convexity. A larger C implies weaker uniform convexity.

The inequality states that the squared norm of the midpoint is less than or equal to the average of the squared norms, *minus* a term that depends on the distance between the vectors. This means that if v and w are far apart, the norm of their midpoint is significantly smaller than the average of their norms. This implies that the unit sphere in a uniformly convex space is "round" and doesn't have any flat spots.

3. Inequality (1.3): This inequality extends (1.2) to any point on the line segment between v and w, parameterized by t ∈ [0, 1]. It's derived by recursively applying (1.2) and using the continuity of the norm. The interpretation is similar: the squared norm of a point on the line segment is bounded above by a weighted average of the squared norms of the endpoints, minus a term that penalizes the distance between the endpoints.

4. Comparison with Ordinary Convexity: The inequality ||(1-t)v + tw||² ≤ (1-t)||v||² + t||w||² follows from the ordinary convexity of the squared norm. Inequality (1.3) strengthens this by adding the penalty term, highlighting the stronger convexity property of uniformly convex spaces.

In summary: Uniform convexity is a geometric property of normed spaces that implies a stronger form of convexity than the standard definition. It ensures that the unit sphere is "round" and that the norm behaves in a well-behaved manner, which has important implications in various areas of analysis, including optimization and approximation theory. The inequalities (1.2) and (1.3) provide quantitative measures of this stronger convexity.