Polygon of maximum area contained in compact, convex subset of R 2 ?Let H be a compact, convex subset of R 2 .. For a given m ≥ 3 ,, let P m be a polygon of maximum area which contained in H and has atmost m sides. Then A r e a ( P m ) A r e a ( H ) ≥ m 2 π sin ⁡ ( 2 π m ) and equality holds if and only if H is an ellipse.

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Polygon of maximum area contained in compact, convex subset of             R        2  ?Let H be a compact, convex subset of             R        2    .. For a given   m  ≥  3  ,, let       P    m   be a polygon of maximum area which contained in H and has atmost m sides. Then                     A        r        e        a        (                  P          m                )                    A        r        e        a        (        H        )              ≥            m              2        π              sin  ⁡      (                            2          π                m              )   and equality holds if and only if H is an ellipse.

Sanaa Hudson · Accepted Answer

Step 1I think even a bound   ≥      ϵ    m   has some value. I will provide an       ϵ    3  .Consider H compact convex body. Take   Δ      A    1        A    2        A    3    of largest area. The points       A    i   are on the boundary. Because we cannot enlarge the area by moving       A    1  , there exists a supporting line to H through       A    1   that is parallel to       A    2        A    3  . Similarly for the other vertices. Thus we can inscribe H in a larger triangle   Δ      A    1    ′        A    2    ′        A    3    ′   for which the       A    i  &#039;s are the midpoints. Therefore   Area  ⁡  Δ      A    1        A    2        A    3    ≥      1    4    Area  ⁡  H.Step 2Notice that if H were the ellipse tangent to the sides of the larger triangle at the midpoints       A    i  &#039;s then we would have the exact bound stated ( the figure would be affinely equivalent to a circle inscribed in a equilateral triangle).

Let H be a compact, convex subset of mathbb{R}^2. For a given m geq 3, let P_m be a polygon of maximum area which contained in H and has atmost m sides. Then (Area(Pm))/(Area(H)) geq m/(2pi)sin((2pi)/m) and equality holds if and only if H is an ellipse.

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