strictly convex space is a normed vector space (X, || ||) for which the closed unit ball is a strictly convex set. Put another way, a strictly convex...

enclosing a strictly convex set of points Strictly convex set, a set whose interior contains the line between any two points Strictly convex space, a normed...

properties. For instance, a strictly convex function on an open set has no more than one minimum. Even in infinite-dimensional spaces, under suitable additional...

that a convex set in a real or complex topological vector space is path-connected (and therefore also connected). A set C is strictly convex if every...

coefficients. It follows that convex cones are convex sets. The definition of a convex cone makes sense in a vector space over any ordered field, although...

In mathematics, uniformly convex spaces (or uniformly rotund spaces) are common examples of reflexive Banach spaces. The concept of uniform convexity...

said to be strictly less than an element b, if a ≤ b and a ≠ b . {\displaystyle a\neq b.} For example, { x } {\displaystyle \{x\}} is strictly less than...

\}}} A (strictly) quasiconvex function has (strictly) convex lower contour sets, while a (strictly) quasiconcave function has (strictly) convex upper contour...

polygon is convex if every line that does not contain any edge intersects the polygon in at most two points. A convex polygon is strictly convex if no line...

does not admit a strictly coarser Hausdorff, locally convex topology. For that reason, the study of sequences begins by finding a strict linear subspace...

{\displaystyle {\log }\circ f} is convex, and Strictly logarithmically convex if log ∘ f {\displaystyle {\log }\circ f} is strictly convex. Here we interpret log...

typically not Banach spaces. A Fréchet space X {\displaystyle X} is defined to be a locally convex metrizable topological vector space (TVS) that is complete...

a\}} are convex sets. A differentiable function f is (strictly) concave on an interval if and only if its derivative function f ′ is (strictly) monotonically...

n} -dimensional Euclidean space R n {\displaystyle \mathbb {R} ^{n}} . Most texts use the term "polytope" for a bounded convex polytope, and the word "polyhedron"...

x_{n}\}}.} The LogSumExp function is convex, and is strictly increasing everywhere in its domain. It is not strictly convex, since it is affine (linear plus...

\end{cases}}} The convex conjugate and Legendre transform of the exponential function agree except that the domain of the convex conjugate is strictly larger as...

locally convex. Other well-known examples of TVSs include Banach spaces, Hilbert spaces and Sobolev spaces. Many topological vector spaces are spaces of functions...

n-dimensional space. That is, the points that are not incident to the hyperplane are partitioned into two convex sets (i.e., half-spaces), such that any...

geodesically convex subset of M. A function f : C → R {\displaystyle f:C\to \mathbf {R} } is said to be a (strictly) geodesically convex function if the...

concept called strictly decreasing (also decreasing). A function with either property is called strictly monotone. Functions that are strictly monotone are...

≤) is replaced by the strict inequality then f {\displaystyle f} is called strictly convex. Convex functions are related to convex sets. Specifically, the...

In mathematics, an LF-space, also written (LF)-space, is a topological vector space (TVS) X that is a locally convex inductive limit of a countable inductive...

the pseudoconvex domain.: 49  Strongly pseudoconvex and strictly pseudoconvex (i.e. 1-convex and 1-complete) are often used interchangeably, see Lempert...

compact convex sets in locally convex topological vector spaces (TVSs). Krein–Milman theorem—A compact convex subset of a Hausdorff locally convex topological...

and the characteristic of convexity are measures of "how convex" the unit ball in a Banach space is. In some sense, the modulus of convexity has the same...

Stanisław Ulam in response to a question raised by Stefan Banach. For strictly convex spaces the result is true, and easy, even for isometries which are not...

include the closed convex curves (the boundaries of bounded convex sets), the smooth curves that are convex, and the strictly convex curves, which have...

real Banach space is reflexive if and only if every pair of non-empty disjoint closed convex subsets, one of which is bounded, can be strictly separated...

a topological vector space (not necessarily locally convex or Hausdorff) over the real or complex numbers. Then the open convex subsets of X {\displaystyle...

Convex optimization is a subfield of mathematical optimization that studies the problem of minimizing convex functions over convex sets (or, equivalently...