**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Conical combination

Summary

Given a finite number of vectors in a real vector space, a conical combination, conical sum, or weighted sum of these vectors is a vector of the form
where are non-negative real numbers.
The name derives from the fact that a conical sum of vectors defines a cone (possibly in a lower-dimensional subspace).
The set of all conical combinations for a given set S is called the conical hull of S and denoted cone(S) or coni(S). That is,
By taking k = 0, it follows the zero vector (origin) belongs to all conical hulls (since the summation becomes an empty sum).
The conical hull of a set S is a convex set. In fact, it is the intersection of all convex cones containing S plus the origin. If S is a compact set (in particular, when it is a finite non-empty set of points), then the condition "plus the origin" is unnecessary.
If we discard the origin, we can divide all coefficients by their sum to see that a conical combination is a convex combination scaled by a positive factor.
Therefore, "conical combinations" and "conical hulls" are in fact "convex conical combinations" and "convex conical hulls" respectively. Moreover, the above remark about dividing the coefficients while discarding the origin implies that the conical combinations and hulls may be considered as convex combinations and convex hulls in the projective space.
While the convex hull of a compact set is also a compact set, this is not so for the conical hull; first of all, the latter one is unbounded. Moreover, it is not even necessarily a closed set: a counterexample is a sphere passing through the origin, with the conical hull being an open half-space plus the origin. However, if S is a non-empty convex compact set which does not contain the origin, then the convex conical hull of S is a closed set.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Related courses (32)

Ontological neighbourhood

L'objectif du cours est d'introduire les notions de base de l'algèbre linéaire et ses applications.

STAY A LITTLE LONGER étudie les potentialités du bâti existant. Les outils de représentations du projet de transformation - Existant/Noir, Démolition/Jaune, Nouveau/Rouge -structureront l'exploration

STAY A LITTLE LONGER étudie les potentialités du bâti existant. Les outils de représentations du projet de transformation - Existant/Noir, Démolition/Jaune, Nouveau/Rouge -structureront l'exploration

Related units (4)

Related concepts (8)

Related people (39)

Related publications (218)

Related lectures (67)

, , , , , , , , ,

Affine combination

In mathematics, an affine combination of x1, ..., xn is a linear combination such that Here, x1, ..., xn can be elements (vectors) of a vector space over a field K, and the coefficients are elements of K. The elements x1, ..., xn can also be points of a Euclidean space, and, more generally, of an affine space over a field K. In this case the are elements of K (or for a Euclidean space), and the affine combination is also a point. See for the definition in this case.

Affine hull

In mathematics, the affine hull or affine span of a set S in Euclidean space Rn is the smallest affine set containing S, or equivalently, the intersection of all affine sets containing S. Here, an affine set may be defined as the translation of a vector subspace. The affine hull aff(S) of S is the set of all affine combinations of elements of S, that is, The affine hull of the empty set is the empty set. The affine hull of a singleton (a set made of one single element) is the singleton itself.

Convex cone

In linear algebra, a cone—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under positive scalar multiplication; that is, C is a cone if implies for every positive scalar s. When the scalars are real numbers, or belong to an ordered field, one generally calls a cone a subset of a vector space that is closed under multiplication by a positive scalar.

Linear Combinations: Vectors and Matrices

Explores linear combinations of vectors and matrices in Rn, demonstrating geometric interpretations and matrix operations.

Linear Equations: Vectors and Matrices

Covers linear equations, vectors, and matrices, exploring their fundamental concepts and applications.

Counting: Structures and Algorithms

Explores the logic behind proofs, structures, and the scalability of solutions in counting and algorithms.

Matthieu Wyart, Carolina Brito Carvalho dos Santos

We study the glass transition by exploring a broad class of kinetic rules that can significantly modify the normal dynamics of supercooled liquids while maintaining thermal equilibrium. Beyond the usual dynamics of liquids, this class includes dynamics in ...

We study the statistical mechanics and the equilibrium dynamics of a system of classical Heisenberg spins with frustrated interactions on a d -dimensional simple hypercubic lattice, in the limit of infinite dimensionality d -> infinity . In the analysis we ...

The present invention relates to systems and methods for predicting a prognosis of the neuropsychological and/or neuropsychiatric status in a subject based on reports of Minor Hallucination (MH) events in combination with electrophysiological data of the s ...

2024