Real Translates

Definition

The translate of $S \subset \R ^n$ by the vector $a \in \R ^n$ is the set

\[ \Set*{z \in \R ^n}{\exists x \in S \text{ such that } z = x + a}. \]

Notation

We often use the abbreviated notation $S + a$ for the translate of $S$ by $a$. It is sometimes also convenient to extend set-builder notation and write

\[ S + a = \Set{x + a}{x \in M}. \]

The right hand side is slick notation for the definition given above.

Sums and differences

The sum (or Minkowski sum) of two sets $S, T \subset \R ^n$ is the set

\[ \Set{z \in \R ^n}{(\exists x \in S)(\exists y \in T)(z = x + y)}. \]

Likewise, the difference (or Minkowski difference) of two sets $S, T \subset \R ^n$ is the set

\[ \Set{z \in \R ^n}{(\exists x \in S)(\exists y \in T)(z = x - y)}. \]

Notation

We denote the sum of $S$ and $T$ by $S + T$, and the difference by $S - T$.¹ We often use the slick notation

\[ \Set{x + y}{x \in S, y \in T} \text{ and } \Set{x - y}{x \in S, y \in T}, \]

for these two sets. Notice that in this notation

\[ \set{a} + B = a + B \]

Scaled sets

Given a set $A \subset \R ^n$ and a $\lambda \in \R $, the set which is $A$ scaled by (or scaled set, scaling) is

\[ \Set{z \in \R ^n}{(\exists x \in A)(z = \lambda x)} \]

We often denote this set by $\lambda A$. As before, we often use the slick notation

\[ \lambda A = \Set{\lambda a}{a \in A} \]

The set $(-1)A$ is denoted $-A$

Homothetic sets

A set $A$ is homothetic to a set $B$ if there is $x \in \R ^n$ and $\lambda \neq 0$ so taht

\[ A = x + \lambda B \]

If $\lambda > 0$, $A$ is positively homothetic to $B$.

This second notation unfortunately conflicts with our notation for set differences. Future editions will correct. ↩︎