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 $B$.
132 questions
2 votes
2 answers
76 views
Minkowski sum of the set of irrational with an open interval is open?
"If $A$ and $B$ are subsets of the set of real numbers where either $A$ or $B$ is open, then the Minkowski sum of $A$ and $B$ is open". I am failing to see how it can be true, as any real ...
0 votes
0 answers
71 views
Sums of a quadratic-generated odd sequence and coverage of the even integers
For $\alpha > 0$, let $$ A(\alpha) = \{ R_{\text{up-odd}}((\alpha x)^2 + \alpha x + 1) : x \in \mathbb{Z}_{\ge 0} \}, $$ where $R_{\text{up-odd}}(t)$ means: take the ceiling $\lceil t\rceil$; if it ...
- 1,403
0 votes
0 answers
33 views
Number of vertices of the convex hull of a full Minkowki sum of n vectors in d dimensions whose sum is zero.
Disclaimer : I'm not very good at maths and I just happen to stumble on this problem during my PhD for a "fun side quest". Hi, A bit of context, I'm working on a kind of vector control, in ...
- 1
2 votes
1 answer
152 views
Is there an infinite set of integers whose pairwise sums are all distinct perfect squares? [duplicate]
Let $S \subset \mathbb{Z}$ be a set of integers such that for every pair of distinct elements $a, b \in S$, the sum $a + b$ is a perfect square, and no two such sums are equal. Does there exist an ...
- 3,613
3 votes
0 answers
89 views
Does there exist an subset $X\subsetneq\mathbb{R}_{> 0}$ which is uncountable in every interval and closed under addition?
Does there exist an subset $X\subsetneq\mathbb{R}_{> 0}$ which is uncountable in every interval and closed under addition? *Closed under addition means that $x_1\in X,\ x_2\in X\implies x_1+x_2\in ...
- 25.2k
2 votes
0 answers
47 views
Support of a infinite sum of independent random variables is equal to the closure of the sum of the supports
Ok, so I know that for $X_n$ independent random variables $$\def\supp{\operatorname{supp}} \supp \left( \sum_{n\leq N} X_n \right) = \overline{\sum_{n\leq N} \supp X_n}.$$ Here, supp denotes the ...
1 vote
0 answers
46 views
Find best shape of given area that minimizes area of its Minkowski sum with unit disc
We want to find what plane shape or collection of shapes of given total area minimizes area of its Minkowski sum (https://en.wikipedia.org/wiki/Minkowski_addition) with unit disc. Minkowski sum of a ...
- 824
1 vote
1 answer
91 views
What is the largest by area "capsule" shape inside ellipse? [closed]
Define "capsule" shape as rectangle with half-circles attached to its opposite sides. Another definition for "capsule" will be a Minkowski sum (https://en.wikipedia.org/wiki/...
- 824