K und L seien konvexe Teilmengen des Vektorraumes E. Zeige:...

$K$ und $L$ seien konvexe Teilmengen des Vektorraumes $E$. Zeige: a) $K+L:=\left|x+y: x \in K, y \in L_{i}^{\prime} . z+L:=\right| z+y: y \in L_{i}$ und $\lambda K:=; \lambda x: x \in K_{i}$ sind konvex $(z \in E, \lambda \in R$ beliebig). b) Für $\lambda, \mu \geqslant 0$ ist $(\lambda+\mu) K=\lambda K+\mu K$. c) Sind $x_{1}, \ldots, x_{n}$ Elemente von $K$ und $\lambda_{1} \ldots \ldots \lambda_{n}$ nichtnegative Zahlen mit $\lambda_{1}+\cdots+\lambda_{n}=1$, so liegt $\lambda_{1} x_{1}+\cdots+\lambda_{n} x_{n}$ wieder in $K$. Hinwe is: Induktion.

$K$ und $L$ seien konvexe Teilmengen des Vektorraumes $E$. Zeige:
a) $K+L:=\left|x+y: x \in K, y \in L_{i}^{\prime} . z+L:=\right| z+y: y \in L_{i}$ und $\lambda K:=; \lambda x: x \in K_{i}$ sind konvex $(z \in E, \lambda \in R$ beliebig).
b) Für $\lambda, \mu \geqslant 0$ ist $(\lambda+\mu) K=\lambda K+\mu K$.
c) Sind $x_{1}, \ldots, x_{n}$ Elemente von $K$ und $\lambda_{1} \ldots \ldots \lambda_{n}$ nichtnegative Zahlen mit $\lambda_{1}+\cdots+\lambda_{n}=1$, so liegt $\lambda_{1} x_{1}+\cdots+\lambda_{n} x_{n}$ wieder in $K$. Hinwe is: Induktion.

Key Concepts

Convex Combinations

A convex combination is a linear combination of points where all coefficients are nonnegative and sum to one. Convex combinations are critical in defining the structure of convex sets because they guarantee that any such combination of points within a convex set also lies within the set. This is a foundational tool in proving various properties of convex sets.

Scalar Multiplication

Scalar multiplication in the context of sets involves multiplying each element of a set by a scalar. For convex sets, when every element is multiplied by a nonnegative scalar, the resulting set remains convex. This property is essential in linear algebra and convex analysis, particularly in understanding and preserving the shape characteristics under scaling.

Mathematical Induction

Mathematical induction is a proof technique used to establish that a proposition holds for all natural numbers. In the context of convex combinations, induction is often used to extend results from combinations of two points (the basic definition of convexity) to combinations of any finite number of points, thereby generalizing the concept.

Convex Sets

A convex set in a vector space is one where, for any two points within the set, every point on the straight line segment connecting these two points is also contained in the set. This property is fundamental in understanding geometric constructions, optimization, and various areas of analysis since it ensures no 'holes' or 'indentations' exist in the set.

Minkowski Sum

The Minkowski sum of two sets is an operation where each element of one set is added to each element of the other set. When the operands are convex sets, their Minkowski sum is also convex. This concept is significant in geometric operations, optimization problems, and in studying properties of convex bodies.

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