An integrally convex set is the discrete geometry analogue of the concept of convex set in geometry. A subset X of the integer grid Z n {\displaystyle...

geometry, a set of points is convex if it contains every line segment between two points in the set. For example, a solid cube is a convex set, but anything...

In geometry, the convex hull, convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined...

function is convex if its epigraph (the set of points on or above the graph of the function) is a convex set. In simple terms, a convex function graph...

In mathematics, convex geometry is the branch of geometry studying convex sets, mainly in Euclidean space. Convex sets occur naturally in many areas: computational...

is the convex set of probability distributions, as linear combinations preserve neither nonnegativity nor affinity (i.e., having total integral one). A...

A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n {\displaystyle n} -dimensional...

360°. This question was first posed, for convex regions, by Sōichi Kakeya (1917). The minimum area for convex sets is achieved by an equilateral triangle...

fixed-point theorems often require a convex set. The analogue of this property for discrete sets is an integrally-convex set. A fixed point of a discrete function...

In integral geometry (otherwise called geometric probability theory), Hadwiger's theorem characterises the valuations on convex bodies in R n . {\displaystyle...

mathematician Johan Jensen, relates the value of a convex function of an integral to the integral of the convex function. It was proved by Jensen in 1906, building...

of functional analysis and convex analysis concerned with measures which have support on the extreme points of a convex set C. Roughly speaking, every...

an integral polytope is a convex polytope whose vertices all have integer Cartesian coordinates. That is, it is a polytope that equals the convex hull...

and the convex hull of a finite set of points is a polyhedron. Many common families of polyhedra, such as cubes and pyramids, are convex. Convex polyhedra...

the Minkowski sum with a vector subtraction. If the two convex shapes intersect, the resulting set will contain the origin. A − B = { a − b | a ∈ A ,   b...

Examples of convex curves include the convex polygons, the boundaries of convex sets, and the graphs of convex functions. Important subclasses of convex curves...

is compact), but it is most useful for convex bodies (that is bodies, whose corresponding set is a convex set). The mean width of a line segment L is...

The Shapley–Folkman lemma is a result in convex geometry that describes the Minkowski addition of sets in a vector space. The lemma may be intuitively...

logarithmically-convex. A Reinhardt domain D is called logarithmically convex if the image λ ( D ∗ ) {\displaystyle \lambda (D^{*})} of the set D ∗ = { z =...

 Theorem 2.3.23  The concepts of convex sets and convex components are used in a proof that every totally ordered set endowed with the order topology is...

down, convex upwards, convex cap, or upper convex. A real-valued function f {\displaystyle f} on an interval (or, more generally, a convex set in vector...

such as the relaxation of the set cover problem discussed earlier, form a polytope that strictly contains the convex hull and has vertices other than...

the injective tensor product of the locally convex topological vector spaces (TVSs) X and Y. An integral linear operator is a continuous linear operator...

forming a convex envelope of the feasible set. However, to capture non-convex solutions, alternative aggregation operators like the Choquet integral can be...

{\mathcal {P}}(Y)} the set of all its convex and compact subsets. Let φ : X → K {\displaystyle \varphi :X\to {\mathcal {K}}} be a convex and compact valued...

vertex arrangements of the convex regular polygons. In general, for any natural number n, there are n-pointed non-convex regular polygonal stars with...

(convex hull) - If a point x of Rd lies in the convex hull of a set P, there is a subset of P with d+1 or fewer points such that x lies in its convex hull...

in different overlapping sets of objects being called polytopes. They represent different approaches to generalizing the convex polytopes to include other...

In mathematics, Minkowski's theorem is the statement that every convex set in R n {\displaystyle \mathbb {R} ^{n}} which is symmetric with respect to the...

ideal in a commutative ring. In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset. In formal languages,...