Intereting Posts

Prove continuity on a function at every irrational point and discontinuity at every rational point.
Prove $\lim_{x\rightarrow 0}\cos (x)=1$ with the epsilon-delta definition of limits
show that $\int_{0}^{\infty} \frac {\sin^3(x)}{x^3}dx=\frac{3\pi}{8}$
Are there countably many infinities?
Functional equation $P(X)=P(1-X)$ for polynomials
Joint probability distribution (over unit circle)
A common term for $a_n=\begin{cases} 2a_{n-1} & \text{if } n\ \text{ is even, }\\ 2a_{n-1}+1 & \text{if } n\ \text{ is odd. } \end{cases}$
Semilocal commutative ring with two or three maximal ideals
Hahn-Banach theorem (second geometric form) exercise
Confused by inductive proof of associative law
Finite Series $\sum_{k=1}^{n-1}\frac1{1-\cos(\frac{2k\pi}{n})}$
I don't understand the 'idea' behind the method of characteristics
Find remainder of $F_n$ when divided by $5$
What's the explanation for these (infinitely many?) Ramanujan-type identities?
Factorise the number $5^{2015} – 1$ into three positive factors such that each is greater than $5^{200}$

If for two convex closed sets $S_1$ and $S_2$, the Minkowski sum is a Euclidean ball then can $S_1$ and $S_2$ be anything other than Euclidean balls themselves. I suspect they can be but I haven’t found a counterexample. I don’t have experience with Minkowski sums so any help will be appreciated.

Thanks!

- Is every monotone map the gradient of a convex function?
- Is every convex function on an open interval continuous?
- Number of ways to separate $n$ points in the plane
- Prove that in triangle $ABC$,$\cos^2A+\cos^2B+\cos^2C\geq\frac{3}{4}$
- Sufficient condition for a function to be affine
- Test if point is in convex hull of $n$ points

- Conditions for augmenting a collection of sets so that the new sets are small but the hull is large?
- Hessian Related convex optimization question
- Pointwise supremum of a convex function collection
- Is there a unique saddle value for a convex/concave optimization?
- How to derive the proximal operator of the Euclidian norm?
- Continuity of a convex function
- Hausdorff metric and convex hull
- Two fundamental questions about convexity of a function (number1)
- A real differentiable function is convex if and only if its derivative is monotonically increasing
- How to prove convexity?

This is almost certainly false. The following animation shows two convex shapes (with outlines shown in red and green) whose Minkowski sum is a disk of radius 3 (with outline shown in blue). The green shape is an ellipse with major and minor radii 1 and 1/2, which uniquely determines the red shape.

I do not have a proof that the red shape is convex, but it shouldn’t be too hard to check.

Incidentally, here is the Mathematica code I used to produce this animation:

`MyPlot = ParametricPlot[{3*{Cos[t], Sin[t]},`

With[{u = ArcTan[-Sin[t], Cos[t]/2]},

3*{Sin[u], -Cos[u]} - {Cos[t], Sin[t]/2}]},

{t, 0, 2 Pi}];

myframes =

Table[With[{u = ArcTan[-Sin[t], Cos[t]/2]},

With[{pt = 3*{Sin[u], -Cos[u]} - {Cos[t], Sin[t]/2}},

Show[MyPlot,

ParametricPlot[pt + {Cos[r], Sin[r]/2}, {r, 0, 2 Pi},

PlotStyle -> Darker[Green]],

Graphics[{PointSize[Large], Point[pt]}]]]], {t, 0, 2 Pi - Pi/20,

Pi/20}]; ListAnimate[myframes]

**Edit:** Here is a simpler solution using two congruent shapes. The boundary of each shape is the union of two circular arcs, each of which is congruent to 1/4 of the blue circle.

Here’s mine. Done before I saw Jim’s solution (honest). But after seeing his, I animated mine, too (using Maple).

Two copies of the Reuleaux triangle

http://en.wikipedia.org/wiki/Reuleaux_triangle

same size, one rotated by 180 degrees from the other.

For $\mu$ a Borel positive measure on the sphere $S^{n-1}$, consider a continuous Minkowski sum of segments

$$ K_\mu = \int_{S^{n-1}} [0,\theta] \, \mathrm{d}\mu (\theta) .$$

The set $K_\mu$ is convex (it could be defined by its support function, then the integral becomes a usual one). Now observe that (1) $K_{\mu+\nu} = K_\mu + K_\nu$ (2) by rotation invariance, $K_\mu$ is a Euclidean ball if $\mu$ is the uniform measure (3) there are many ways to write the uniform measure as a sum of positive measures.

- If $f \circ g$ is onto then $f$ is onto and if $f \circ g$ is one-to-one then $g$ is one-to-one
- the exact value of $\displaystyle\sum_{n=2}^\infty\arcsin{\left(\dfrac{\sqrt{n}-\sqrt{n-1}}{\sqrt{n^2-1}}\right)}$
- Can a finite axiomatization of PA be expressed in a finitely axiomatizable first order set theory?
- Calculating prime numbers
- How can we find and categorize the subgroups of $\mathbb{R}$?
- Curves with constant curvature and constant torsion
- Can anyone clarify how a diverging sequence can have cluster points?
- If p $\equiv$ 3 (mod 4) with p prime, prove -1 is a non-quadratic residue modulo p.
- Structure Descriptions (GAP) in semigroups
- Explicit homeomorphism between open and closed rational intervals?
- Finding minimal form — Velleman exercise 1.5.7a
- $ \lim x^2 = a^2$ as $x$ goes to $a$
- Are there nontrivial vector spaces with finitely many elements?
- Finding the value of $\int_0^{\pi/2} \frac{dt}{1+(\tan(x))^{\sqrt{2}}}$
- Convergence of sequence: $f(n+1) = f(n) + \frac{f(n)^2}{n(n+1)}$