Questions tagged [sumset]
For questions regarding sumsets such as $A+B$, the set of all sums of one element from $A$ and the other from $B$.
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Regarding scaling in sumsets
Let $A$ be a finite subset of $\mathbb{N}$. We denote the set $\{a_1 +a_2: a_1, a_2\in A\}$ as $2A$. We call the quantity $\sigma[A]:= |2A|/|A|$ as the doubling constant of $A$, and this constant can ...
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A Question on the Brunn-Minkowski inequality
It is a direct consequence of the Brunn-Minkowski inequality that
\begin{equation}
|A\oplus B| - \Big(\sqrt{|A|}+\sqrt{|B|}\Big)^2 \geq |A\oplus\tilde{B}| - \Big(\sqrt{|A|}+\sqrt{|\tilde{B}|}\Big)^...
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answer
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Affine Combinations and Span
I was reading a bit of convex analysis and came across this problem.
Let $S$ be convex. Let $A$ be the set of finite affine combinations of points in $S$ (i.e. finite linear combinations whose weights ...
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Sum of closed and compact set in a TVS
I am trying to prove: $A$ compact, $B$ closed $\Rightarrow A+B = \{a+b | a\in A, b\in B\}$ closed (exercise in Rudin's Functional Analysis), where $A$ and $B$ are subsets of a topological vector space ...
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What is the maximum range of a convex finite additive 2-basis of cardinality k?
Conjecture:
Given any $d \in \mathbb{Z}_{\geq 2}$ and $k=2d-2$, we have \begin{align*}
\max \{ n : (\exists &f \in \{ \mathbb{Z}_{\geq 0} \to \mathbb{Z}_{\geq 0} \})\\ &[((\forall i \in \...
- 115
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Generating restricted finite additive $2$-bases from doubly-eager bit-strings
A bit-string is any finite sequence of $1$s and $0$s. For example, $1011011$, $1011010$, and $000110$ are bit-strings.
In this post, I will refer to bit-strings as strings, to be concise.
I now ...
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Whether $\sup(\sum\limits_{i=1}^{\infty}X_i)$ is equal to $\sum\limits_{i=1}^{\infty}(\sup X_i)$
We have $\sup(A+B)=\sup(A)+\sup(B) $.Thus, we have $\sup(\sum\limits_{i=1}^{n}X_i)=\sum\limits_{i=1}^{n}(\sup X_{i})$ for every finite integer $n\in\mathbb{N}$. However, what about the set sequence? ...
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How many subsets $S$ of integer interval $[0,n]$ such that $n, n-1 \not \in S+S$?
Conjecture:
Given any $n \in \mathbb{Z}_{\geq 0}$, we have $$|\{S : (S \subseteq [0,n]) \land (n, n-1 \not \in S+S)\}| = F(n+2),$$ where $F$, the sequence of Fibonacci numbers, is given by $F(j) = F(...
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Convexity of minkowski space when two triangles collide
I am working on a Python program to show the collision of two triangles by using Minkowski difference. I am subtracting each point from one triangle from every other point on the other triangle. The ...
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Is this Minkowski Sum result correct?
Is this Minkowski Sum result correct?
I expected a filled shape as it happens when the two polygons don't overlap (longer translation vector).
Full discussion here: https://github.com/AngusJohnson/...
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Minkowski sum of the intersection of a closed and an open set with a compact set
Consider $\mathbb R^n$ with the usual topology and the Borel sigma-algebra. Let $A$ be open and $B$ be closed sets, respectively, in $\mathbb R^n$. Let $C$ be a compact set. Is the set $(A \cap B) \...
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answer
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What's the difference between the Minkowski difference of $A$ and $B$ and the Minkowski sum of $A$ and $-B$?
In the book Computational Geometry, Algorithms and Applications from de Berg, van Kreveld, Overmars and schwarzkopf, I read the following in chapter 13.3 on Minkowski sums:
Sometimes $ P \oplus(-R(...
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Is the Minkowski sum of a compact set and a closed set necessarily closed?
Why is it that if we have a compact set $X$ and a closed set $Y$ then the Minkowski sum $X+Y$ is necessarily closed?
Sorry for keep asking questions about the Minkowski sum, I am trying to figure out ...
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Minkowski sum of disks in 3D
Suppose we have a set of disks in $\Bbb R^3$. These disks are neither parallel nor perpendicular to each other. In general, is it possible to formulate (or write an equation for) the object ...
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The Minkowski sum of two convex sets is convex
Let $A$ and $B$ be two convex subsets in $\mathbb{R}^n$.
Define a set $C$ given by
$$C = A + B = \{a + b : a \in A \mbox{ and } b \in B\}.$$
Is $C$ a convex set?